They are both correct depending on the context. They are both short forms of longer sentences.
Someone says to you, " I don't like football."
I don't like football either = Me either = I don't like football too.
Here, you are repeating the statement for emphasis.
It is a parallel statement that emphasizes your enthusiastic agreement with the other person.
You would never say, "I don't like it neither", because double negatives are not used in English.
Neither you nor I like football = me neither = more formally, Neither do I. or Nor do I.
Here, you are stating that you BOTH don't like football.
This is considered correct formal usage and is naturally more polite because you include the other person in your statement.