I'm not a strictly grammatical myself, but I seem to have a sensitivity what's correct or incorrect.
The whole neither/either argument can get confusing, but whenever someone says "Me either", they're put into the same category as those who use 'then' for 'than', and 'there' for 'they're' (or 'there' for 'their'), etc.
But one is nonsense, so if you want to make a negative impression, go for it. Someday you'll make a negative impression on the wrong person, affecting your entire life. But young folks don't usually worry about that until it's too late.