"me neither" was wrong. :-(
I would use that phrase on its own, but not as part of this sentence. So if someone said 'I am not good at X' then I might reply 'Me neither'. But as part of this phrase it does not seem right.
I see that now!
Because we're British. :P
I put "me neither" too - and "nor me"!
What is wrong with 'and nor do I'?