"...," said an older version of me.
But a native speaker of English — which I am not — replaced the me with myself. Can someone tell me which one is correct and why?
To replace "me" with "myself" is to try to make a deliberately (I presume) inelegant construction more elegant by resorting to wrong grammar. It's the worst solution.
If you were writing a serious piece you would re-write it completely as, for example, "I said in a previous phase of my life". But if you are writing frivolously, then keep the frivolity intact. "Me" sounds more comical than "myself", and is also correct grammar.
In one song from The Sound of Music, the lyricist has deliberately resorted to bad grammar in pursuit of character portrayal: "I have confidence in me". In this case, "myself" would be correct, but wisely the lyricist overrode that with a something that is more in keeping with the character that sings the song. If I am right in thinking that you are also writing in a frivolous mood, then stick with "me".
The reflexive pronoun is required when the subject and complement of the verb (or preposition) are the one and the same person. But in the sentence "Quote," said an older version of me, the subject is "an older version of me" and the complement is "Quote". Hence the 'rule' does not apply and me is grammatical in this context.
However, it is common to encounter the use of a reflexive when the context does not require it. Garner in the prescriptive grammar Modern American Usage (p533) describes such uses as "untriggered reflexives", which are "thought somehow to be modest, as if the reference were less direct."
The descriptive Cambridge Grammar of the English Language (p1495) refers to the same usage as "override reflexives" and notes: "Much the most common override is 1st person myself." The CGEL further states:
"Quote," said an older version of me/myself is a context where some native speakers would prefer the override reflexive to the non-reflexive pronoun.
myself is used when both the subject of the clause and the object (of a verb or preposition) are the speaker, e.g. I cooked dinner for myself.
me is used when the subject is someone other than the speaker, e.g. My girlfriend cooked dinner for me.
In your example, there's no explicit subject, because you used the passive mode. But the implied subject is yourself, so myself would be correct.