He didn't have the courage to go against what was expected. He instead was complicit and didn't fight against the system that was sending him to a war he did not want to go to. He would go and face the bullets that would come his way rather than face the looks on people's faces that would be given at him and think he's a coward for not fighting for his country. His family would look at him and think that they did not raise a "man". They would think he ran away and hid rather than do his duty.
He was too brainwashed to be able to live with the humiliation that he did not fight. He wasn't strong enough to be able to live that down. He wasn't strong enough to stand up against the system, not courageous enough to be an activist, he didn't have enough will to defy his country. He would do what he was told like a good little sheep, and for that, he believed himself a coward.
Personally, I would consider myself a different type of coward, I would not go to war unless I fundamentally believed in the cause, but I would be an activist, I would stay and protest. I wish he would have swam.