I don't like white wine.
We're talking about subjects here, so naturally the pronoun should be "I". The use of "me" would only make sense to me if "neither" was a postposition. According to a dictionary it can be all kinds of things but not a preposition.
- Why are we using "me" here?
- What part of speech is neither?
- Is this sentence a shortened version of something that explains the weird grammar?