When Henry returns to the regiment, he explains that he received a bullet wound to the head so he won't have to admit that he deserted in battle.

I think Henry made a wise decision to lie. Although most of the soldiers ran as well, it is understood that all must participate in a performance of bravery. Henry's honesty would have destroyed the facade that the soldiers were trying to keep. This would have caused self-recrimination and pain that would be all projected onto Henry.