It's one of those things English speakers generally use correctly without ever thinking about how they're using it. Either means "one or the other" and neither means "not one and not the other". On the face of it, they're opposites, but in usage terms, not always.
In your example, "me neither" means "you say not you and I say not me too". The clumsiness of my explanation shows why people say "me neither" as shorthand.
"Me either" doesn't make a lot of sense unless you preface it with the modifier "not". Saying "not me either" is generally accepted as meaning "me neither" even though, strictly, it doesn't have to mean that.
"'Me either" or "Me neither?"
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It's one of those things English speakers generally use correctly without ever thinking about how they're using it. Either means "one or the other" and neither means "not one and not the other". On the face of it, they're opposites, but in usage terms, not always.
In your example, "me neither" means "you say not you and I say not me too". The clumsiness of my explanation shows why people say "me neither" as shorthand.
"Me either" doesn't make a lot of sense unless you preface it with the modifier "not". Saying "not me either" is generally accepted as meaning "me neither" even though, strictly, it doesn't have to mean that.