You can not hunt with a slingshot in Connecticut. The regulations prohibit it. He can use an air rifle, but not a slingshot. A squirrel falls under the definition of quadruped.

26-66-1. Behavior and actions of hunters

(a) The method of taking wild birds and wild quadrupeds by hunting shall be restricted to firearms, high velocity air guns using a single ball or pellet type projectile, and compound, long, or recurved bow. In designated areas, by special permit, the commissioner may allow the use of modified versions of air guns or archery implements to achieve wildlife management objectives. No arrow or projectile which is coated with or contains any drugs, poison or known tranquilizing substance may be used or possessed while engaged in archery hunting.

Sec. 26-1. Definitions. Words and terms used in this chapter shall be construed as follows:

(13) "Quadruped" means any four-legged animal which is ferae naturae or wild by nature, although such animal may be enclosed and considered a pet or semidomesticated, but shall exclude purely domesticated animals.