I'm an not an English teacher. I teach French to Chinese speakers but I happen to teach them English once in a while. I studied the use of 'both' with them so I can tell you a little about it, if I may...
Both means 'the 2 of them' and becomes 'neither' in a negative context.
Both Mary and Judy are secretaires.
= both of them are secretary.
(Mary is a secretary and so is Judy)
Neither Mary nor Judy is a secretary.
= neither of them is a secretary.
(Mary isn't a secretary ans neither is Judy)
Hope this helps!