For me, (Southern UK), a hypothetical question about an action that has already been completed should refer to what he should have done, rather than what he actually did. Thus, if he did give up, you would use the first sentence. If he did not give up, you would use the second sentence.
An actual (rather than hypothetical) question should refer to the action that you think is the correct one, expecting an answer of yes. Thus, a parent might ask
Have you done your homework?
In the Northern UK (I can personally vouch for Scotland and Yorkshire), an additional not is often added to questions without the same overtone of suggesting the answer that you want to hear:
Have you not done your homework?