Nearly all the ground squirrels I've ever shot have been Belding's ground squirrels, generally called sage rats, but it doesn't sound like those are what the OP is talking about. Have shot literally thousands upon thousands of sage rats over the past 45 years. Started shooting them when I was around 10 with a Win model 74 22lr my grandpa let me use, I still have it. Over the years I've shot them with a variety of different 22's but once the 17HMR came out and started using one of those I hardly ever shoot the 22's.
If most of the shooting is in the 50-ish yard range it's pretty hard to beat a 22lr but once the shooting starts reaching 100 or 100+ yds then the 17 HMR really shines. I got a 17 WSM a few years ago and that has made shooting sage rats at much further distances and still with a rimfire, a lot of fun. Was down in Oregon a couple years ago and had a little group of them on a mound out at 350 yds and was able to connect pretty regularly on them.
Rimfires are always fun to shoot, especially on live targets like ground squirrels, but screw a suppressor on the end and the fun meter stays on high all day long. There's no way around the sonic crack but a good rimfire suppressor, even my 17 WSM, makes me giggle like a little girl everytime I pull the trigger and the damn thing sounds like a pellet gun.