I haven't seen any style guide that recommends using double periods (full stops) inside and outside a close quotation mark, which appears to be your central question. A number of style guides disagree about the preferability of placing the period inside or outside the close quotation mark (either as a blanket rule or in response to situational variables), but none that I know of votes for placing them inside and outside the close quotation mark.
The Chicago Manual of Style, fifteenth edition (2003), which reflects standard U.S. publishing practice on this particular issue, spreads its advice over a couple of sections:
CLOSING QUOTATION MARKS IN RELATION TO OTHER PUNCTUATION
6.8 Periods and commas. Periods and commas precede closing quotation marks, whether double or single. This is a traditional style, in use well before the first edition of this manual (1906). ...
6.122 No double period. When an expression that takes a period ends a sentence, no additional follows [cross reference omitted]. ...
In your example sentence
The statement previously appeared in the document as "Use eye protection.".
the period inside the quotation mark effectively ends the sentence, so Chicago 6.122 indicates that there should be no second period outside the close quotation mark.
Words into Type, third edition (1974) is even more succinct:
Use [of the period] with other marks. ... With quotation marks. When the period is used with closing quotation marks, place it inside [cross reference omitted].
Use [of quotation marks] with other marks. Set quotation marks outside of periods and commas.
Both of these Words into Type guidelines militate against putting a second period outside the close quotation mark in your example sentence.
The standard British style for handling end punctuation in relation to material within quotation marks is more nuanced than (and I think undeniably superior to) the standard U.S. style. Here is the general advice in The Oxford Guide to Style (2002):
5.13.2 Relative placing [of quotation marks] with other punctuation
Except when the matter is quoted for semantic or bibliographic scrutiny, the relationship in British practice between quotation marks and other marks or punctuation is according to sense. While the rules are somewhat lengthy to state in full, the common-sense approach is to do nothing that changes the meaning of the quotation or renders it confusing to read.
When the punctuation mark is not part of he quoted material, as in the case of single words and phrases, place it outside the closing mark. Usually only one mark of terminal punctuation is needed. When the quoted matter is a complete sentence or question, its terminal punctuation falls within the closing quotation mark, and is not duplicated by another mark outside the quotation mark: [examples omitted]
It follows under Oxford's guidelines that, since you are dealing with a quotation containing a complete sentence (with end punctuation)—
Use eye protection.
—you should put the quotation mark outside the period and not duplicate the period outside the quotation mark.
The upshot of all this is that Chicago, Words into Type, and Oxford agree that the punctuation of your sentence should be
The statement previously appeared in the document as "Use eye protection."