You can hear people say it but no, you should say "me neither". I'd say the difference is between correct and incorrect. To use "either", you need to have a "not" or "don't" or some form of negation. Like "I don't either".

As far as I know, 'either' means one of two choices. ===> I want to have either a dog or a cat. 'Neither' means none of the choices available. ===> I like neither spiders nor snakes.

You will hear them used interchangeably by native speakers, so it's safe to say that both ways are fine. However, if you would like a more detailed explanation, here is some more info... Both expressions are shortened versions of the same statement. Take this sentence for example, "I do Not want to go Either." In English, there technically should be a negation (here, it is the word "not") in order to use the word "either." So, the word "neither" is an abbreviation for the words "not either." We never say the words "not either" this way though. Therefore, "me either" and "me neither" are both short for "I do not want to either" (or some variation of this sentence, i.e. "I do not have any either")

Alright... since there are not already enough opinions on this question... ; ) Both are entirely intelligible spoken English heard regularly. However the correct 2 word response is 'nor I'. The correct long versions are: Nor do I go there. I don't go there either. Neither do I go there. As Luke suggested, these last 2 probably shorten to 'me either' and 'me neither' respectively.