The only concern I have in using the .38 Special for squirrels is that on high angle shots the heavier spent bullet is more dangerous coming down.
I use my Marlin 24" Cowboy Limited with light .38 Spl. loads which are the minimum charge which will exit the barrel reliably. You must use very soft swaged or cast bullets, with a secure crimp adequate to hold the bullet against compression of the magazine spring, and a fast burning powder which provides reliable ignition.
I use either the Remington factory swaged 158-gr. SWC lead bullets or the Saeco #358 160-gr. FN cast of soft alloy such as plumber's lead or not harder than 1:20 tin/lead and 2.7 grs. of Bullseye in .38 Special cases, I load these unsized, tumble lubed with Lee Liquid Alox and WSP primers. Rounds are heavily crimped using the Lee Factory Crimp Die, which also profile sizes the loaded round and sizes the bullet inside the case.
DO NOT USE JACKETED BULLETS - or hard cast ones because you will stick one in the barrel!
From a 24" barrel these make no more noise than standard velocity .22 LR. Velocity is about 800 f.p.s. and they shoot 2" groups at 50 yards.
I haven't had as good luck making very light loads in .357 cases as there is too much air space in the case.