Reference: http://learnersdictionary.com/qa/The-Difference-Between-Neither-and-Either https://www.grammar.com/either_vs._neither Either and neither can be used in a few different ways, and to mean a few different things. In most cases, neither means "not either." Specifically, either means "one or the other" and neither means "not one or the other". A great way to remember the difference between these two words has to do with the first letter of the word neither. Neither creates a negation and starts with the letter “N.” Neither means not one of the given options. If you can remember this trick, you’ll be all set. In informal speech either and neither are used in the phrase me either or me neither to say that you agree with a negative statement someone has made. These two phrases mean exactly the same thing. The following example sentences show this use. "I don't like this outfit." "Me either." [meaning: I also don’t like this outfit] "I wasn't happy with the way the movie ended." "Me neither." [meaning: I also wasn't happy with the way the movie ended]

This is a question of grammar. "Me either" is not grammatically correct. Here they are in proper usage. It is difficult to explain. "I'm not going to the movie." "Me neither." OR "I'm not going to the movie." "I'm not going either." I think it's that "neither" is a negative response and using "either" is a more positive response, but I'm not completely sure. If someone can provide a more technical explanation, please do.

Neither we use in negative sentences and either in positive

thank you very much.

Informal American usage, not too grammatical. You'll find a lot of informal usage, British and American, which does not follow the grammar taught in grammar books :)