Would the climax be, if the story was from Schmuel's perspective, being taken to the concentration camp or when he went to the gas chamber? If the climax was when he goes to the gas chamber, how is there any time for falling action to occur?

Shmuel and Bruno's entering of the gas chamber does indeed provide the climax to The Boy in The Striped Pajamas. In a conventional plot structure, the climax is followed by the falling action. And here the falling action consists of Bruno's family trying to come to terms with the tragedy of their loss. This is only possible because the story is written from a third-person perspective, which allows the author to provide us with an objective insight into the characters, their behavior, and their motivations. As the author intended that Shmuel and Bruno would meet their deaths together it would've been impossible to have a falling action after this tragic climax had their story been told from either of the boys' first-person perspective.