The first symbol is the snow; Tub is introduced while standing in the snow, waiting for the others. Throughout the story, the snow is both an obstacle to the hunters and a representation of their minds; it is both hostile and uncaring, and entirely too large to deal with alone.
The second symbol is the "No Hunting" sign.
They followed the tracks into the woods. The deer had gone over a fence half buried in drifting snow. A no hunting sign was nailed to the top of one of the posts.
(Wolff, "Hunters in the Snow," classicshorts.com)
This sign is the boundary where the story takes its darker turn. Had the men given up, Kenny wouldn't have played the joke that resulting in his shooting; the sign warns them back and is ignored.
The third symbol is the dog. It serves the purpose of a dog, running and barking, and apparently it is infirm enough for the farmer to want it put down. It represents the innocence that is lost by the men on the hunting trip; the dog expected love or playing from humans, not death, and so too does the hunting trip remove any vestiges of innocence that Tub and Frank have built for themselves.