In Chapter 3 Candy allows Carlson to put his dog out of its misery and shoot it. Carlson takes the time to explain where the dog should be shot to ensure the quickest death and least amount of suffering. Later in the chapter Candy states that he should have done this, that "you shoot your own dog." The shooting of Candy's dog in Chapter 3 foreshadow's events to come later in the book.
Just as the dog is Candy's responsibility, Lennie is George's responsibility. As Lennie becomes more violent, George understands that Lennie will need to be killed. George understands that this is something that he needs to do to own up to this responsibility as well as to ensure that Lennie is taken care of in the least painful manner possible.
Candy says he should have shot his old dog himself rather than having allowed Carlson to kill it because he feels that he should have been man enough to have realized that the poor dog suffered and not kept him for selfish reasons.
In the time setting of Steinbeck's Of Mice and Men, there was little that the dispossessed men of the Great Depression could call their own. Many had lost their farms, their homes, and their jobs; further, they owned little more than what they carried in a bedroll. So, for Candy, the ownership of a dog set him apart from the others; he had a creature to love and a being who loved him. Then, of course, when he realized the loss of this beloved creature would leave him alone, he could not bear to put the suffering dog down. Thus, he is devastated when Carlson shoots his pet.
However, after Candy feels that he has a share in Lennie and George's plan to own a small farm, Candy's sense of fraternity enriches him just as his sense of camaraderie with his dog did. So, he is able to become more objective about the dog's physical condition and can say that the dog was decrepit and that he should have stopped the pet's suffering himself.
That Candy is strengthened by being included in the dream of a farm is evinced in his quick comment to George after he is included in George and Lennie's plans: "George . . . I ought to of shot that dog myself. . . . I shouldn't ought to of let no stranger shoot my dog."
Carlson feels that Candy's dog should be shot because it is old and useless. Carlson petitions Candy to shoot his dog and says that he should put it out of its misery. However, Candy is reluctant to shoot the old dog for several reasons. Candy tells Carlson that he has had the dog for such a long time that he couldn't bring himself to shoot the animal. Candy has a soft spot for his old dog and can identify with it. Similar to his dog, Candy is also old and almost useless on the farm. He sympathizes with the old dog's situation and refuses to kill the animal that he loves. After Slim gives a reasonable argument for why the dog should die, Candy feels helpless and allows Carlson to shoot his dog. After Carlson shoots his dog, Candy feels guilty and comments that he shouldn't have let a stranger kill his dog. Candy feels like he should have been a stronger man and shot the dog himself. The dog was Candy's old friend and it was only right that he should be the one to end its life. This scene helps explain why George ends Lennie's life at the end of the story when the mob is closing in on him.