A squirrel gun or squirrel rifle is a small-bore rifle suitable for hunting squirrels and other small game. While it is hardly a formal designation, traditionally a squirrel gun would generally be about .35 caliber, while a .40 might be called a turkey gun, a .45 a deer rifle, and a .50 a bear rifle. In modern terminology, a .22 rifle is likely to be the most popular gun for squirrel hunters. 'Squirrel gun' is approximately the American South equivalent of the garden gun.
While most dictionaries (and Google) don't seem to have caught on yet, this is not usually what is meant by a squirrel gun. In my experience, 'squirrel gun' is primarily used to refer to a gun -- any gun -- that is under-powered for the job at hand. This appears to have started in WWII. British and American tanks, in the early years, were severely outgunned by the German Panzer and Tiger 1 tanks. The Allies' anti-tank guns were often ironically referred to as squirrel guns or squirrel rifles by the troops, referring both the the comparatively calmer pop of their firing and their ineffectiveness against enemy tanks at a distance.