When somebody makes a negative statement, and I agree with them, you could use these expressions. However, I think "me either" is more common in the US. Although "me neither" is commonly heard, it is probably not really grammatically correct. I always say, "Neither do I." or "Nor do I." I was always taught that this was 'better' English. The choice of expression is probably linked to region, class, and level of formality... (At least in the UK...) There is also the question of pronunciation: neither can be 'nei-' like 'knee', or 'nei-' like 'nigh' or n+ 'eye'. Both are correct, but (again) I was told that the latter (the 2nd one) was 'better' pronunciation. "You pays your money and you takes your choice" (UK idiom = there is little choice between two alternatives)...

I'm from the U.S. and I would consider ME EITHER to be incorrect. People might say it anyway, but ME NEITHER is much more common.

Both of them mean that you are agreeing with something someone said, however, “me neither” is the grammatically correct one.