The expression "free, white, and twenty-one" goes back at least a bit farther then 1854, when the first through third editions of Alone (Richmond, Virginia: 1854), by Marion Harland (pen name of Mary Virginia Terhune) seem to have been published. That occurrence is the earliest instance of the expression mentioned in FumbleFingers's answer.
The practical sense of the expression is evident in the following longer wording, which embeds the gist of the expression. From "An Act to Amend the Acts Relating to the Town of Georgetown," in Acts of the General Assembly of the Commonwealth of Kentucky: Passed at the Session Which Was Begun and Held in the City of Frankfort, on Saturday the 31st of December, 1853, and Ended Friday the 10th of March, 1854 (1854):
Be it enacted by the General Assembly of the Commonwealth of Kentucky, That section fourth an act to amend and reduce into one the several acts in relation to the town of Georgetown, approved March 1st, 1847, be so amended as to read as follows to-wit: That said Chairman and Trustees shall be elected annually, on the first Saturday in January, by the free white inhabitants of said town over the age of twenty one years, who shall have been bona fide residents of said town for six months next preceding the election, and who shall have paid their poll tax to said town chargeable to them for the year next preceding the election at which they claim to vote; also by the free white male inhabitants of Scott county, over the age of twenty one years, who shall be owners of real property lying within the limits of said town. ... Approved March 1, 1854.
The language in this act regarding "free white male inhabitants of said town" and "of Scott county" was the same in section 4 of the 1847 act; the amended language of 1854 simply added the requirement about paying a poll tax. In fact, the wording "free white male inhabitants over the age of twenty one years" appears multiple times in the 1847 Kentucky statutes.
Much earlier, the same prerequisites are mentioned in "South Carolina" in A System of Geography; or, A Descriptive, Historical, and Philosophical View of the Several Quarters of the World (Glasgow, 1805):
In South Carolina, as in other American States, the legislative power is vested in a general assembly, consisting of a senate and a house of representatives. The number of senators is 35; that of representatives 124. The representatives are chosen biennially. To be qualified for this office, a person must be a free white man, 21 years of age; must have been an inhabitant of the state three years, and, if he reside in the district for which he is chosen, he must have a freehold clear of debt to the amount of 150 sterling. ... To be entitled to the privilege of voting for members of the legislative body, a person must be a free white man, 21 years of age, must have been an inhabitant of the state two years, and must have been, for six months preceding the election, possessed of a freehold of 50 acres of land, or a lot in a town.
The earliest instance of the exact expression "free, white and twenty-one" that I have found is from "The Right Key," an article originally printed in the Fredericksburg [Virginia] Recorder (September 3, 1845), reprinted in Niles' National Register (Baltimore, Maryland: September 13, 1845):
Our appeal is to the people! We are satisfied that editors may importune the legislature till the crack of doom, without one particle of effect. The experiment has been tried frequently and ardently, but yet the humiliating truth will still stare us in the face that 58,000 of our people—"free, white and 21,”—cannot write their name, or read it when in print! Will the legislature try to remedy this monstrous evil? No; not until each member is made to feel that his official existence depends upon his action on this subject.
The fact that wording "free, white and 21" appears in quotation marks suggests that the writer was invoking a formulation that was already (in 1845) a familiar phrase—presumably one used to identify the prerequisites for having come of age and entered into the full rights of citizenship in Virginia at that time.
The wording "free, white and 21 years of age" also appears in A Conservative, "Thoughts on the Virginia Convention," in The Jefferson Monument Magazine (Charlottesville, Virginia: November 1850):
The next great change which is proposed [for the Virginia state constitution], is to have universal suffrage. If it does occur, we will have universal degradation with it. Under the present system, Free-holders, House-keepers and Lease-holders are voters, whose property may be as little as $25 or a house 12 feet square. Now we confidently assert that any man who is incapable of obtaining a vote under these conditions, is unworthy of it. If he does not possess that much mental, moral and physical energy, his vote would degrade the candidate, the office, and , if possible, himself. He is unworthy the title of citizen, and should not participate in the government. Suppose, for an instance, that we gave every man who was free, white and 21 years of age, a right to vote--what would be the result? The suffrages of the idle, indolent and ignorant would be as valuable, and in many cases counteract those of industrious, active, and learned.
From "Free Speech in Virginia" in the Norfolk[Virginia] Herald (circa November 1859) reprinted in Anti-Slavery Tracts, series 2, numbers 1–14 (1860):
The other [instance of a person committing the "indictable offence, punishable by fine and imprisonment, to give utterance to Abolition language and sentiments" in the state of Virginia] was that of a resident of Ferry Point, opposite the city, John Fletcher by name, who came from Washington City some five years ago. On Tuesday last, in the grocery store of his neighbor, Mr. James P. Jones, in the presence of ten creditable witnesses, while in conversation about the Harper's Ferry affair, “he avowed himself an Abolitionist, and asserted that there were many in Norfolk and Portsmouth, but that they were afraid to say so; but he was free, white and twenty-one, and had no hesitation in declaring that if he had five thousand dollars, he would give one-half of it for the release or rescue of John Brown."
And from "Communicated," in the Alexandria [Virginia] Gazette (February 9, 1860):
At a meeting of several citizens of Alexandria, held on the 8th instant, composed of men, "free, white, and 21," of sufficient intelligence to know their legal privileges, and duties of citizens, and of sufficient independence to maintain them, and not dependent, in any way, upon the present Administration, or professing to be its apologists or supporters, it was unanimously Resolved, That ...
Nevertheless, the expression appears in some early contexts in which voter qualifications are clearly not the intended implication. From The Undercurrent, serialized in Scott's Monthly Magazine (Atlanta, Georgia: November 1868):
The morning lessons of the children are done, their books are ranged neatly on the opposite shelves, they dismissed for their daily walk with the nurse, and Lina is employing the moments of their absence in jottings in her journal. We will look over her shoulder and read as she writes:
"Today I complete my twenty-first year.
'Free, white and twenty-one
My woman's life scarce begun.'
Yet how old and worn I am beginning to feel beneath the pressure of these daily cares, trials and disappointments.
Revisiting the wording in Alone (1854), we see that "free, white and twenty-one" is again being "sung" by a female character:
"Never fear," said Emma [to Ida]. "Your Richmond party could consume it [all of the food being prepared] in a week. How many are there?"
"Let me see! Arthur, Carry, and my pet—three—Mr. and Mrs. Dana, three children, and Charley—nine. They will be here to-morrow night—Ellen, Morris, Monday or Tuesday. I have invited Anna Talbot and Josephine,but do not expect them. Then for Tuesday evening—from the neighbourhood—Dr. Hall and lady, and a friend, who shall be nameless," pinching Emma's cheek—"the Strattons, Kingtons, Frenches,—and oh! I gave Charley a carte-blanche to ask any of my Richmond acquaintances—and all for what? To hear that Miss Ida Ross is—"
"Free, white, and twenty-one" sang Emma, cheerily.
So if Emma in 1854 and Lina in 1868 are both either singing or quoting an actual song, it may be that "Free, white, and twenty-one" are lyrics from it.
In the antebellum U.S. South, the privileges associated with being "free, white, and twenty-one"—and male and a property owner, and (if applicable) a poll tax payer—were widely understood to represent a legal status—that of a person who possessed full citizenship: the rights to vote, to hold (some) political offices, to serve on juries, to enter into binding contracts, to marry without parental permission, to serve in a militia or other military organization. The formulation seems to have been common enough to support its adoption as a idiomatic equivalent of "independent adult with full rights as a citizen."
Bartlett Whiting, Modern Proverbs and Proverbial Sayings (1989) cites instances of "free, white and twenty-one" as a proverbial phrase going back to 1932, in Cecil Gregg, The Body Behind the Bar: A Tale of Inspector Higgins: "She's free, white, and twenty-one." (Oddly enough, Gregg was a British writer and this mystery novel was published in London.) I couldn't find a searchable copy of this novel online to confirm the fuller context in which the expression appears.
As documented elsewhere in this answer, however, the phrase appears as a sort of proverbial expression or descriptive shorthand long before 1932. A typical instance is from "A Fighting Kentuckian Meets Adventure on the Western Front," in The Literary Digest (March 16, 1918):
Of the many thrilling accounts of life in the trenches under fire that have been published in the last year or two, none is more vivid and stirring to the blood than the story of the experience of a hardy son of the Blue Grass State who decided in October, 1915, that being free, white, and twenty-one, it was his manifest duty to strike a blow for democracy. So he traveled in hot haste to Canada and enlisted in the Canadian Grenadier Guards.
However, the expression in something close to its later, aggressively egocentric, unbeholden-to-anyone sense appears by the early 1920s. One instance where the expression seems to be used in a strikingly pejorative way is in J.A. Dickey & E.C. Branson, "How Farm Tenants Live in Mid-State Carolina," in Home and Farm Ownership: North Carolina Club Year-Book, 1921–22 (1923):
- Civic Consciousness. But there are comforting signs of intelligent appreciation of the way-out in Baldwin and Williams townships. The straw ballot taken in the fifty-one tenant households shows that only seven of the ninety-nine voters were opposed to consolidated schools, only three were opposed to cooperative marketing, road bonds, or 'book farming,' only two thought college education a waste of time, and only two considered themselves free, white, and twenty-one and privileged to do as they pleased without regard to morals, law and order. And mark this—of these ninety-nine voters, seventy-two are in the habit of voting regularly.
Similarly—and even earlier, is this instance from Ruthella Bibbins, Mammy 'mongst the Wild Nations of Europe (New York, 1904):
Every now and then, as if to point Manny's remarks, a party of Americans sped by—some with the polished cosmopolitan manner and well-groomed suitably dressed exterior which stamps the cultivated American, the world over. Others there were, with the loud aggressive air, which the Professor says indicates: "Please note I am free, white, and twenty-one." These are they whom one is glad one doesn't know; but, as Mr. Howells says, "gladder still to know they are here, finding out how much they didn't know before."
It thus appears that "free white and twenty-one" was for a considerable amount of time in the first half of the twentieth century a double-edged sword that could emphasize a (white) person's arrival at full and sober maturity or a (white) person's arrogant claim to unchecked autonomy, regardless if the effects exercising that autonomy might have on others.
There is also some suggestion that the expression was associated with rustic or poorly educated people and not with sophisticated urban people. For example, from Francis Smith, Kennedy Square (New York, 1911):
"But the next one [dance] is mine," exclaimed Harry suddenly, examining his own dancing-card. He had not shifted his position a hair's breadth, nor did he intend to—although he had been outwardly polite to the intruder [Willits]
"Yes—they'd all be yours, Harry, if you had your way," this in a thin, dry tone"—but you mustn't forget that Miss Kate's free, white, and twenty-one, and can do as she pleases."
Harry's lips straightened. He did not like Willits's manner an he was somewhat shocked at his expression; it seemed to smack more of the cabin than of the boudoir—especially the boudoir of a princess like his precious Kate.
The farther "free, white, and twenty-one" got from its roots in the Southern U.S. as an encapsulation of the most-favored-citizenship status under law, the less it became about formal rights and responsibilities and the more it became simply a declaration of freedom to do as one pleased. By the time it began appearing in Hollywood movies of the 1930s, it seems to have become a nonregional catch phrase to indicate a headstrong (and sometimes reckless) belief in one's autonomy and self-sufficiency.
As an aside, I note that I grew up on the edge of the Old South (in southeast Texas) in the 1960s and 1970s, and I never heard anyone use the expression "free, white, and twenty-one." (I can't say the same about certain other racially fatuous expressions such as "that's mighty white of you.") So it may be that this catch phrase was already obsolescent—at least in the U.S. South—by the end of the 1940s.