Dear Jackie: I think the replies "Me either" or "Me neither"
are idioms, and so not susceptiable to grammar.
I did find a joke with your friend's version:
'Me neither'

Bob and Joe sat next to each other taking a test. When they
finished, the teacher called them up to the front of the room and said, "Boys, I will have to give both of you a zero on this test"

"W-why?" they wanted to know, though Joe was shifting uncomfortably.

She said, "Your answers were too nearly alike. One of you cheated and the other one let him do it."

"What makes you think we cheated?" Bob asked. "That could
have been a coincidence."

The teacher said, "I might have believed that if it wasn't for the fact that when you came to question #10, Bob wrote in 'I don't know' for the answer, and you, Joe, put 'Me neither'!"