Utilitarianism is not just concerned with happiness; there is a pragmatic viewpoint which determines the greater good for the most amount of people, a type of utilitarianism advocated by William James.
Taken from the pragmatic viewpoint, George's action of killing Lennie in order to prevent him from suffering can, then, be considered utilitarian. In this respect, too, the moral worth of George's action is determined by its outcome. So, since his was a mercy killing of Lennie, it is utilitarian.
George's utilitarianism is only focused on himself and those he cares about. He is not pursuing the happiness of the whole group. If he were, he might have allowed them to take Lennie because that would have made the majority happy.
Instead, we see George acting in ways that will result in the greatest happiness for those he cares about, particularly himself and Lennie. His killing of Lennie is the most important example of this, but we also see it when he gets Lennie out of trouble, as he did in Weed.
So, George's actions are based on what he thinks will bring happiness to himself and those he cares about. He does not care much about the rest of the communities in which he is working.
This is going to be difficult because we are taking a philosophical idea and placing it in a literary context. The situations might be incompatible because the literary context was created in the mind of an artist and the philosophical setting seeks to be a real world construct. In its most crude form of utilitarianism representing "the greatest good for the greatest number," George's killing of Lennie represents an act of utility because the mob of workers believed Lennie to be guilty and wanted him dead. If George would have resisted or defied the wishes of the mob led by Curley and Carlson, then he would have been in violation of the majority's wishes. In a way, it is representative of the shooting of Candy's dog, something that Carlson argued was of no use to anyone. In killing Lennie, the utilitarian might see George as having done what the majority wished. When Slim approaches George at the end and says, "You had to do it," the implication might be that in order to maintain cohesiveness with the will of the majority, George had little choice. I am not entirely certain that this was Steinbeck's intention, but if we are to assess George's actions from a utilitarian point of view, this is what might be gained.