I thought it might be useful to explore this ambiguity a little further, so here goes:
First, my understanding that "himself" refers to Ennis, not Jack, is essentially a minor expansion of the original:
"'Jack, I swear...,' he said, although Jack had never asked him to swear anything and he himself[i.e. Ennis] was not the swearing kind."
The advantage, I think, of this reading is that it keeps the focus on Ennis at this critical moment. We know that Ennis has never been asked by Jack to swear anything, so in doing so now, is not fulfilling some promise he may have made to Jack in the past. We also know that the taciturn Ennis is not given to making solemn pronouncements or oaths. Even so, as Ennis contemplates the two shirts, his own now covering and protecting Jack's, he is so profoundly moved that, probably for the first time in his life, he offers Jack a solemn statement or pronouncement starting "Jack, I swear..."
I feel if "himself" refers to Jack we lose this concentration on Ennis and it is replaced by a not particularly relevant piece of information that Jack wasn't "the swearing kind". So what?
But it's a possible reading. To each his own choice. But undoubtedly a choice has to be made.