The answer to the first part of this question is left up to reader interpretation. The story does not directly tell readers whether Henry drowned on purpose or by accident. Readers are left to infer this answer based on his characterization throughout the story. Personally, I think that Henry killed himself at the end of the story. Before going off to war, he was a very happy and vibrant individual. He loved cruising around in the car and literally swooping ladies off of their feet and twirling them around in the air:

Then my brother Henry did something funny. He went up to the chair and said, "Jump on my shoulders." So she did that, and her hair reached down past his waist, and he started twirling, this way and that, so her hair was flung out from side to side.

"I always wondered what it was like to have long pretty hair," Henry says. Well we laughed.

That Henry never returned from the war. Instead, the narrator gets back a Henry that is jumpy, quiet, and exceptionally closed off.

When he came home, though, Henry was very different, and I'll say this: the change was no good. You could hardly expect him to change for the better, I know. But he was quiet, so quiet, and never comfortable sitting still anywhere but always up and moving around.

It doesn't matter whether you call it "shell shock" or PTSD, the war has severely affected Henry, and he no longer finds joy in any of the things that he previously enjoyed. I truly believe that Henry recognizes this about himself. I believe that he doesn't want to live like that anymore, nor does he want people around him to have to put up with it. I think he chooses to kill himself to end the mental prison that he's in.

In Thomas Wolfe's famous novel he writes, 

...you can't go home again....You can't go back to your family, back home to your childhood, back home to romantic love, back home to a young man's dream of glory and of fame, back home to exile, to escape...back home to lyricism, to singing just for singing's sake, back home to athleticism, to one's youthful ideas....

Henry, like so many soldiers, especially those held as prisoners, cannot go back to what he was before the war; he cannot return to the carefree brother before going to Vietnam where he was made a prisoner of war despite the efforts of Lyman, his brother, who makes every effort to restore Henry to his former self.

Something died inside Henry; he had to shut it off while he was prisoner, and now he cannot turn it back on, even in his own home, or out in the country with Lyman where his emotions are as turbulent as the water he steps into, to cool off after his agitated attempts at frivolity. With the strong current and flooded waters of the river, Henry cannot control his direction, and he is swept away; detached from reality, he realizes he is drowning as he says in a detached voice: "My boots are filling."  With his drowning, then, Lyman "drowns" the red convertible, a symbol of brotherhood. He wants no reminder of the tragedy.