I've heard that "me neither" is incorrect. Instead one should say "neither do I." People definitely say "me neither" conversationally, but is it technically incorrect?
Me neither is idiomatic English. It is a set phrase, as @cindi said. The OED has it as colloquial, originally USA, meaning nor I. Among the many examples from COHA:
The earliest citations, as far as I can tell, are often in the dialogue of characters who do not speak completely standard English, especially before ca. 1930. This pattern no longer seems to hold.
To this American English speaker, Me neither seems perfectly natural. Not formal, but not incorrect either.
It is incorrect in almost every example, for three reasons.
Firstly, Me is the object personal pronoun and is therefore only to be used when the speaker is the object of the verb.
Secondly, Neither is part of the Neither/nor construction (like either/or) and should not be used to indicate the second of two objects in such a construction.
Finally, it is back to front. Neither/nor should precede the subject/object it is modifying.
The grammatically correct phrase is usually "Nor I," which efficiently expresses inclusion with the previous speaker in the following case:
"I have no idea."
"[Neither you] nor I [has any idea]."
If you replace the phrase in the above sentence with "Me neither," the error becomes obvious. The exception to the "Me/I" switch occurs when the preceding speaker is talking of himself/herself as the object of a verb rather than the subject. For example:
"He didn't give me anything."
"Nor [did he give] me [anything]."