Squirrel sniping IS the most enjoyable hunting on the planet - as easy or challenging as you care to make it, and they're easier to pack out than whitetails. More generous bag limits, too.
Handguns: S&W Mod. 41 semi-auto with any accurate HP; any accurate .38/.357 with light loads and a safe backstop.
Rifles: For 30+ years, Anschutz 54 Sporter in .22 Mag., with 'solids' and a 4x Leupold; same in .22 LR with selected HPs (RWS Subsonics), CCI SGBs, or Velocitors; old (like me) Win. 52B Sporter with Lyman 48 and same general loads as the Anschutz; for under 50 yards - Browning .22 Auto in .22 Short, with 4x scopes and HPs. Specialty loads include CCI CB Longs (nearly silent in long barrels and near my set point of aim/impact at 25 yds; head or center-chest shots are effective). I otherwise avoid 'solids' in the long rifle as I've had a few escape wounded and can't justify the experience. Accuracy testing of different available loadings provides confidence for the longer shots.
I haven't tried the .17 HMR, yet, and would like to hear of anyone's experiences with this round on bushy-tailed tree rats.