The other answerers have given you responses which are correct, but they haven't told you the full story. This is a more complete explanation:
'Me neither' is an informal response which can follow any negative comment.
'Neither do I' has the same meaning, but it can only be used in response to a comment in the present simple tense using the auxiliary verb 'don't'.
I don't like broccoli.
Neither do I.
You cannot use 'Neither do I' in any other situation. In other cases, you have to use the appropriate auxiliary verb or modal verb. Here are some examples:
I can't understand this message.
Neither can I.
I haven't done my homework.
Neither have I.
I didn't go to the party.
Neither did I.
I wouldn't buy those shoes.
Neither would I.
So, as you can see, these responses are by no means interchangeable. 'Me neither' is s simple all-purpose response, the equivalent of 'Me too'.
'Neither xxx I' is a construction that has to change according to the grammatical structure of the original sentence.