It's not a matter of "correct"-ness. It's a matter of social class and personal presentation.
In decreasing order of falutation:
- nor do I (archaic; now either ironic or intended to sound classy, or both)
- neither do I (normal in writing, common in speech)
- me either (common in speech, more familiar)
- me neither (common in ingroup speech)
... roughly speaking, of course; I'm no sociolinguist.
This kind of social layering for language variation is completely normal, in every language. To call one mode "correct" would be wrong; appropriateness of speech or writing depends on the speaker/writer's aims, and their judgement of their relationship with their listener/reader (and of course on how accurate their judgement is). What's correct in one situation is often incorrect in another.