Igor,
The word "either" is used in modern English after a negative statement to indicate that a situation is similar to the one already mentioned, or to add additional information using a negative statement.
-If you do not go, he will not go either.
-He didn't like the movie, and she didn't like it either
-It's a problem he can't solve, and I don't think I can either.
- I know a good fast food place. It's not far from here, either.
The informal "me, either" is used in the following way to show that you agree with a negative statement:
"She didn't like the movie." "Me either". It is the equivalent of saying: I also didn't like the movie. or I didn't like the movie either.
Note: You cannot logically say, "I didn't like the movie neither."
More formally you could use "neither":
"She didn't like the movie" "Neither did I".
-She didn't like the movie. Neither did I like the movie.
Another informal possibility is, "She didn't like the movie." "Me neither"