I don't like either X or Y vs. I don't like X or Y either
Solution 1:
I don't like cherry juice or soda.................= a negative statement
I like neither cherry juice nor soda..............= a negative statement
I don't like either.................= a negative statement in response to a question 'Do you like cherry juice or soda?
I don't like cherry juice or soda either.............= as negative statement in agreement with someone who says they don't like cherry juice or soda