I think what you are referring to is the expression "Me neither", which is a informal way of expressing that you share someones negative statement.
Regarding "Either vs Neither", the former basically means "one" and the latter means "none", in other words positive vs negative. "Not...either" would equal "neither":
Sharon: "I don´t like ice cream."
Formal you: "Neither do I." (or "Nor I")
Informal you: "Me neither."
or
"I don´t either."
All have the same meaning: Similar to Sharon, you don´t like ice cream.
Because of the negative "don´t", another negative (neither) is not required in the last example.
"Me too" on the other hand, is used with positive statements.
Helen: "I like ice cream."
Informal you: "Me too."
Formal you: "As do I."