The correct expressions are:
"Neither he or she IS going to the party"
"Neither he or his friends IS going to the party"
"Neither player HAS recovered"
"Neither of the players HAS recovered"
In formal, written English "neither" should be used with a singular verb, because "neither" means no one and indicates a singular.
(In spoken English, however, it is often acceptale if neither is used with a plural verb.)
When there are more than two, "none" should be used instead of "neither", and it should also be used with a singular verb.