I'm not sure where you are in the book, so I don't want to give anything away. Finny's injury ends up being very closely tied to his feelings on the war, and we discover this later on in the novel. But without knowing any specific details, it helps to think of Finny's personality. He is a fun-loving guy who likes to be a leader, the center of all of the action, devising all of the schemes, and forging the way for everyone else. With the war, Finny probably feels the same way; he wants to lead the charge, to be in the middle of it, to be the one who joins up and fights the battles. It is just in his nature to want to be part of the action, any action. So, to be injured, it's got to be a bummer for him. He probably knows that it decreases his chances of getting into the war; because his leg isn't whole and strong, they might not accept him. This would be hard for him to take-he is accepted to everything. He can talk his way into anything, or out of anything. But he knows that he probably can't talk his way into the war, because of his injury. For once, things might be out of his control, and there is nothing he can do about it. Because of this, Finny, instead of being positive about the war and all excited for it, will probably feel bitter towards it. He might even be fearful and apprehensive. If there is a war on, that he might not be able to join, well, then it's a dumb war anyway. To cope, he might have to have a cavalier attitude, and pretend to not care when he really actually does.
I hope that those thoughts help; keep reading, and you will see exactly how Finny does deal with the war. It is an interesting reaction, but it fits with his character, and my thoughts above.