'Should' goes with both verbs. If 'should' did not go with 'have', then it would say 'has' instead of 'have'.
Read it both ways. This is what you are claiming it says:
whoever believes . . . have everlasting life.
But this is what it really says:
whoever believes . . . should . . . have everlasting life.
Say it out loud. 'Whoever have' is not grammatically correct. 'Whoever should have' is grammatically correct.