In British English, we only say 'Me neither'. It's the more informal form of responses such as 'Neither do I' 'Neither have I' 'Neither can I' and so on. 'Me either' sounds wrong to me. I believe that 'Me either' is used in American English, but perhaps an Am Eng speaker could confirm this.

American speaker here: When speaking we also say "me neither."

Depends on where you are from. Some people say "me either" and others say "me neither." I read that "me neither" is the correct way, but both are used.