You are exactly right, FumbleFingers, that "you're not the boss of me" is a childish (or childhood) equivalent of "you're not my boss." In fact, in southeast Texas, where I spent the first 16 years of my life, it was a standard riposte in Childspeak, covering much the same ground as "You're not my mother," but without the demeaning acknowledgment that your mother was in fact the boss of you.
By way of illustrating the difference in core usage, I offer these two exchanges:
Babysitter: It's 8:00—time to put away your toys and get ready for bed.
Five-year-old Sven: You're not my mother!
Nervous ten-year-old goody-two-shoes: If you don't stop playing with those matches, I'm telling!
Ten-year-old Sven: You're not the boss of me!
Although "you're not the boss of me" could be addressed by a child to someone much older—a teenager, say, or even an adult stranger or (in moments of irrational fury) a parent—it was far more frequently used as a response to other children of the same age, or to siblings.
The phrase was pronounced with a marked stress on the first and last words:
You're not the boss of me!
which emphasized (as Ascendant's answer notes) the asserted equivalence in status between "you" and "me." In contrast, "You're not my boss" and "You're not my mother" dedicate their entire focus to the position of "you" in the social hierarchy, while "me" very nearly disappears from consideration. "You're not the boss of me" is a much more effective way to frame the case that the other kid has no business trying to tell you what to do. Those young people really had it figured out.
Very early occurrences of the phrase
A look into the earliest occurrences of the phrase turns up several instances of it from long ago—and from quite distant geographical locations. From "As by Fire," in The Church (London, March 1883):
His sister was going to put her arms around him, but he whirled, and facing her with a very angry face, snapped —
"Let me alone ; you are not the boss of me now, I tell you, and I'm going to do as I please."
From A.C. Stevenson, Unspotted From the World (Chicago, New York, London, 1899):
"Blacksmithing is honorable enough, but ignorance and selfishness are not. You ought to be made to stop going with him, that's all."
"Well, I guess you're not the boss of me."
From Louise Hale, "The Trunk in the Attic," in The Smart Set: A Magazine of Cleverness (New York, May 1912):
Sade says not to write you any bad news so I won't. I am leaving tonight. Twelve years old, and I never rode on a car or a steamboat, so it's time I did—and I can't live in the house with that Johnnie—without you. Mother, I'll die if I can't see you. Johnnie is not the boss of me, is he? But don't you worry, mother dear; he tore the biggest hole in the seat of his Sunday pants. (A nice “man of the house”!) I just am sick to see you.
And from Western Australia, Parliamentary Debates (Perth, 1927) [combined snippets]:
My personal view is that it might be advisable to disband the Routes Advisory Committee, and if necessary its members could be attached to the Traffic Branch. The trouble is that the traffic has too many bosses. I am reminded of a grandson of mine, about three or four years old, who came to visit me, and on one occasion I was chiding him for doing something. He looked up into my face and said, "You are not the boss of me." Even a child objects to too many in control of him, and such a situation arouses in him a spirit of antagonism.
The mysterious upsurge in usage in recent decades
The rise of "You're not the boss of me" in published writing since the 1980s is a mystery to me. I must not be reading the right books, because I haven't noticed it showing up at all frequently in print—and I haven't heard an adult say it, except as a joke, ever. I like the suggestion in Matt E. Эллен's answer that a TV show theme song contributed to its rise since the turn of the century, but clearly it was already gaining popularity during the two decades before that.
Individual Google Books matches for the phrase "not the boss of me" over the years 1970–2008 don't yield any obvious candidates for credit or blame, so if there is an explanation to be had, I don't think it will come from Google Books.