There are more Google matches for the first sentence, but the last one sounds better to me. Which one is correct?
Don't use no double negatives, as a general case. While the use of "not bad, either" is a bit on the informal side, in order to maintain proper logical sense, if you wish to use "neither", you must not use "not."
The food was good. The drink was also good.
The food was good; the drink was not bad, either.
Neither the food nor the drink was bad.