Crooks's American dream is the same as the one that animates the other men. For a brief moment, he becomes genuinely excited about the idea of being part of George's dream of owning his own small farm, where he can live off "the fat of the land." The dream of independence, of being one's own boss, of being able to take a day off when one likes, of being rooted in a community rather than wandering, and of being able to choose one's own companions, has a great appeal to all the ranch hands.
Crooks's desire for it is, I believe, Steinbeck's way of communicating the universality of this deep-rooted dream: it cuts across race, age, and mental or physical ability as a deep and heart-felt desire.
Because of the realities of racism and Crooks disappointments and isolation—he is not even allowed to live in the bunkhouse with the other men—he is the quickest to perceive the futility of the dream, at least for him, and to quickly lapse back into the reality he knows.