A BB gun is capable of killing a squirrel. This depends on the power of the gun, the skill of the hunter and the penetration point. Gray squirrels have a reputation for hardiness, but even .177 caliber, the smallest available for a BB gun, is capable of killing.
If the goal is to kill the squirrel, larger shot is preferred to lighter. This is because very small shot can penetrate the body without hitting any vital organs, resulting in unnecessary suffering on the part of the animal. On the other hand, smaller shot can be used by weaker guns to generate sufficient power to make a quick kill. All hunting with air guns should be done at close range to ensure the shot has sufficient power to kill the target and that the shot is accurate. As BB guns often lack rifling, they are much less accurate at long range.