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:
- "Hast thou a wife?" "No." "Me neither." (Hemingway)
- "I don't understand, Queen," said the Prosecutor finally. "Me neither," groaned Dakin. (Ellery Queen)
- "I don't want to ever get married," Rita said. "Me neither," Toad Tarkington agreed fervently." (Stephen Coonts)
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.