For me, only 'Me neither' is correct. In fact, I was not aware of 'Me either' before I joined italki, and I was surprised to learn that this form existed and was widespread in the USA. 'Me either' sounds very wrong to British ears. As you're from Europe, I presume that you'll have learnt British English. If you say 'Me either' to people who aren't familiar with informal American English, they will probably assume that you are making a non-native error, and may even correct you. In fact, 'Me neither' and 'Me too' are the lazy man's responses. Don't forget that we have a whole set of short-answer responses echoing the auxiliary or modal verb: I don't like x .........Neither do I I can't do x.......Neither can I I won't do x..... Neither will I I'm not doing x.....Neither am I I haven't done x...Neither have I and so on. Or you can just take the easy option and say 'Me neither'. But it might be wise avoid 'Me either' unless you're in the southern USA.

I'm from the southern USA, and I use "me either" all the time. There are probably regional differences but "me either" seems more logical. Consider the following "translation": You do not like it. => I do not like it either. => me either. (This is why I use "me either") You do not like it. => Neither do I (like it). => Neither me => Me neither. (Sounds odd to me.)

Me neither makes the most sense to me

I think a lot of people might use both, but "me neither" is without a doubt the most natural sounding and common way to say it, as it also implies negation.

Of those you mention, "me neither" is most common. However, it's informal and, my impression is that it is not as well established in British English, so a more neutral way to put it is "neither can I".