Yes, it is correct to say 'Me neither'.
It's an easier option than the other type of response which has to follow the grammar of the original sentence. 'Neither do I' is ONLY a response to a comment in the past simple:
I don't understand it.
Neither do I.
Comments in other tenses need the auxiliary appropriate verb form:
Neither did/can/could/have/had/will/would/shall/should/am/was I.
And you do have say the correct one! Native speakers instinctively choose the correct form, but foreign learners might struggle to get it right - so for that reason 'Me too' or 'Me neither' may well be a lazy (but wise) choice.