Subject And Object Pronouns

… be him / he.

Both are correct. You will hear the former much more often.

But we have to use a subject pronoun when the pronoun that refers to the subject follows a form of the verb be is that right or not?

In formal writing, the subject case is used after a linking verb, but in conversation, native speakers invariably choose the object case in sentences like this one:
The winner of the election will be him.

Avoid the issue and say:
He will be the winner of the election.

I'm going through english exam, so I need an explanation about the formal method, any help will be appreciated.

There isn’t much to add to what has already been said. The rule is that a pronoun that is functioning as a subject complement must be in the nominative case.