No. It should be 'Me neither'.

I'd agree with Su.Ki. It's not correct in British English at all. You do hear it a lot in American English though, don't you?

No, it's not correct, and I never use it myself, but I've seen statistics that indicate that it is much more common than the correct form in speech.

Yes, Hmwmp. When people say 'me either', they mean 'me neither', not 'me too'.