Why does X-Ray feel entitled to take what the other boys find?
X-Ray's claim of entitlement does not extend to the other boys and is limited only to what Stanley finds. When Stanley finds a fossilized fish in chapter 11, X-Ray approaches him after Stanley has informed Mr. Pendanski of his discovery. He then asks Stanley to give him whatever he digs up in future. He explains that ever since his arrival a year ago, he has not once gotten a day off for finding something. He informs Stanley that his nickname has nothing to do with his eyesight and is only a reversal of his name (pig latin), which is Rex. He states that he is "too blind to find anything."
X-Ray believes that it is unfair that Stanley should get a day off when he has only been at the camp for a few days, while he has had none. He contends that it would be fair for him to also get a day off. It is for these reasons that X-Ray feels entitled to what Stanley finds.
Stanley does, in fact, later discover a gold tube holder that he gives to X-Ray, albeit somewhat reluctantly. The warden then rewards X-Ray with a double shower, the day off, and some clean clothes while the rest of the boys have to work much harder to see if they can find anything else of value. A while later, Stanley correctly concludes that:
They weren't just digging to "build character." They were definitely looking for something.
He and Zero discover, almost at the end of the novel, exactly what they were supposed to find: a casket of treasure that the warden believed had been hidden by her not too distant ancestors.
Life is not so pleasant for the boys at Camp Green Lake in Holes. However, X-Ray can help. He's the leader of the pack, the one who assigns the other boys nicknames, and the one who often will protect them when they get into trouble--or could get into trouble. He will often, for example, try to deflect the Warden's attention from certain boys or smooth over what might become arguments. For example, when Stanley, without meaning to, angers another boy when he first arrives, X-Ray smooths the situation over before it can escalate.
On the other hand, because he is the leader of the pack, and because of the way he sometimes protects the others, he feels he is entitled to what the other boys find. In the end, X-Ray feels he should take care of himself.