Here are some rules to remember when using BOTH, NEITHER or EITHER.
Both = this AND that
Either = this OR that
Neither = NOT this and NOT that.
Both is used with 'and'
e.g.: "Emma and Megan both went to the party."
We don't usually use this with a negative sentence, but use neither instead.
e.g.: "Both of us don't swim regularly." WRONG.
"Neither of us swim regularly." MUCH BETTER!
Either is used with 'or'
e.g. : "Do you want either chocolate or crisps?"
We often use neither with 'nor', although this is quite formal.
e.g. : "Neither Caroline nor Marguerite worked for EC during the world cup."
Be careful not to use neither with another negative
e.g. : "I don’t want neither chocolate nor crisps." WRONG. We cannot have a double negative!
(from English channel, ECE)