A: How are you? B: Not bad. A: Me ______. (What should be filled in the blank? Either, or too? Since there is a "not" in the sentence, I think it could be "either." What would a native speaker fill in the blank? The correct answer from the text book is " too" ) Many thanks for any reply to my question!!
I would say 'Me, too', ignoring any possible logic involved. However, 'not bad' is not really a negative statement; it is a fixed phrase:

not bad:
a.tolerably good; not without merit

Also, 'not so bad', 'not too bad'.
Same here would also be ok.
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Completely wrong, also neither is wrong. Those words are used for making comparisons!