I don't like white wine. Me neither. 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. ...
I was reading Agatha Christie’s The Murder of Roger Ackroyd and noticed following sentence: About those boots of Ralph Paton’s. It is really in old (old as in early or mid twentieth century) ...
The activity we engaged in was good for she and I. or The activity we engaged in was good for us both. or The activity we engaged in was good for her and me.