What details led to Papá being labeled as Inu by other Japanese and how did he react to allegations?

"Inu" is the Japanese word for dog, but in this context it means an informer or traitor. It's the name given to Jeanne's father by some women at Terminal Island. They refer to him this way because he was released from Fort Lincoln earlier than the other men. It is inferred from this that Papa received special treatment from the authorities in return for supplying them with information. Yet in reality, Papa did no such thing. Indeed, his behavior during his lengthy interrogation at Fort Lincoln was actually much more ambiguous and evasive than the vicious gossip would suggest.
For Papa, this false accusation is a source of great shame. So much so that he refuses to venture outside the crowded barracks for months. In Japanese culture, loss of face is considered deeply shameful, and so one can understand why Jeanne's father would rather hide away than show his face in public, even though the rumors about him are completely false.