The bear and squirrel symbolizes Jimmy's and Alison's immature role-play game to escape from their socially and emotionally violent class differences. Jimmy and Alison are bear and squirrel living in a jungle where steel traps lie all around them, symbolizing the pitfalls and dangers in life. At the end of the play, Jimmy says to Alison: "We'll be together in our bear's cave, and our squirrel's drey, and we'll live on honey, and nuts. . .And we'll sing songs about ourselves--about warm trees and snug caves, and lying in the sun. And you'll keep those big eyes on my fur, and help me keep my claws in order, because I'm a bit of a soppy, scruffy sort of a bear. And I'll see that you keep that sleek, busy tail glistening as it should, because you're a very beautiful squirrel, but you're none too bright either, so we've got to be careful." (Osborne 96)
In the end, Jimmy and Alison admit their need for one another's constant support, saying they are "very timid little animals" who are afraid of the "steel cruel traps" all around them. Alison's steel trap took away her child and any future motherhood, and the play equates this with Jimmy's loss of his father at ten, when he learned about "love...betrayal...and death."
All that bmadnick said is perfect but I'd like to add one last thing. This reference to animals also has to do with Man as a being in the universe. All modern drama put the stress on that element. Man is no longer honourable. That seemly being that used to be grand and even something of divine in Classical and Renaissance drama is lost. In modern drama, man reeks (have a look at Beckett's Waiting For Godot) or leads a disgusting and base life (O'Neill's A long day's journey into night) This intensive reference to animals (apparent from the first stage directions) is there to show that man fell. He is no longer noble, he is peer to animals. This is a direct resulant of the fall of the anthropocentric world and Darwinism. Since Man is not seated at the top, in the center, he is merely an animal. This relegation underlines his unability to make sound judgement or to undertake any wise action at all, hence, he is utterly unable to act on his life.
In the end, the bear and the squirrel symbolize the fall of man and the deconstuction of the whole humanity which is no longer viable as a noble race, if not viable at all, since the two characters are not even of one and the same race. That explains why Alison could not bear an infant to term and why there is no real communication between her and Jimmy. The loss of his humanity seems to say that man is dying.