If you have looked out in the yard and seen squirrels chasing each other, they may not be fighting or playing. The chase may be part of the courtship process.
Squirrels chase each other for a variety of reasons. Young squirrels chase each other for fun while older squirrels chase each other to assert dominance in the mating process. When a squirrel population gets too crowded then they will chase each other to protect the limited food supply.