I would want to argue that the main parallel between these two characters is the way that they present soldiers as fully developed, sensitive human individuals who share a similar distaste to the moral quagmire in which they find themselves in and are just innocent youngsters, with their own lives, interests and loves.
In "On the Rainy River," for example, the narrator presents himself as a very tender young man in all of his emotional complexity. His attempt to escape to Canada, and therefore avoid the draft, is explored in its fullness, especially in the reasons we are given for him deciding in the end to stay and not cross the border. Note how he explains his decision to stay:
All those eys on me--the town, the whole universe--and I couldn't risk the embarrassmet. It was as if there were an audience to my life, that swirl of faces along the river, and in my head I could hear people screaming at me. Traitor! they yelled. Turncoat! Pussy! I felt myself blush. I couldn't tolerate it. I couldn't endure the mockery, or the disgrace, or the patriotic ridicule. Even in my imagination, the shore just twenty yards away, I couldn't make myself be brave. It had nothing to do with morality. Embarrassment, that's all it was.
Such descriptions present us with the full emotional complexity of the narrator, and indicate how he is a sensitive young man caught up in forces beyond his control that he feels he must comply with.
In the same way, in "The Man I Killed," the narrator creates a background for the life of the Vietnamese soldier that he kills, painting him as another sensitive young man. Note the doubt that the narrator gives this soldier as he grows up and how he finds himself unable to fight those who bully him:
He could not make himself fight them. He often wanted to, but he was afraid, and this increased his shame. If he could not fight little boys, he thought, how could he ever become a soldier and fight the Americans with their airplanes and helicopets and bombs? It did not seems possible. In the presence of his father and uncles, he pretended to look forward to doing his patriotic duty, which was also a privilege, but at night he prayed with his mother that hte war might end soon. Beyond anything else, he was afraid of disgracing himself and hterefore his family and village.
Note the similarities between these two characters. O'Brien seems to create a kind of double in the life and background that he gives his victim. Above all, what motivates both of them to be involved in the war is the fear of shame and embarrassment. Both are painted as complex, sensitive characters that find themselves in a war that they are unsure about but nevertheless feel obliged to participate in.