If you're shooting squirrels you are hunting them in the eyes of the law. If they are out of season or you don't have the right license expect to pay a big fine at the very least. And some states have gone mental about how they enforce hunting laws. Both Ohio and Kentucky have handed out fines in the $50,000 range
for hunting deer out of season. I doubt you'd see that much of a fine for squirrels but with the way tree huggers are these days I wouldn't put anything out of the realm of possibility. I would certainly check with a game warden before carrying a second gun and hunting different game no matter what state you're in.
As far as what .22 ammo is "best" that can cover a lot of turf. What's best for hunting is not what's best for target shooting. And every gun is different in terms of accuracy and ammo. What's good in one gun won't be good in another most likely. Guns are finicky things.
If you want to kill squirrels I would use a full power round. I made the mistake of shooting a squirrel with a CCI "Quiet" round recently and the squirrel ran off after I shot it and didn't die for about a day. I like things to be DRT. I was shooting near the house and didn't want to alarm my new neighbors but my landlord wanted to get rid of the squirrels invading the house I am now renting so I shot one. That doesn't qualify as hunting BTW because the squirrel is a true nuisance.
For extreme accuracy I would start with Wolf MT and maybe CCI SV. Both are sub-sonic so they may not cycle an automatic but both make all my semi-autos work. But again every gun is different. For target type accuracy people generally shoot something like Tenex or Federal Gold Medal Ultra Match. Both are over $200 a brick. CCI SV is $30 a brick and Wolf is about $55 a brick or so. Maybe more. And prices are going up again. And every lot is different. One lot of Wolf may shoot great and the next one shoot terrible in your rifle.
Rifles are funny things and that's especially true for rimfire rifles. I'd probably want one of the CCI hunting rounds for what you're going to do. Stingers, Velocitors, etc. are great for hunting but keep in mind that some semi-autos don't like hyper velocity ammo. It can tear up your rifle. Marlins have a buffer that gets broken by hyper velocity ammo.
The quest for the perfect ammo is the biggest test of how a person learns to shoot extremely well IMO. You have to learn to recognize what is working well and when you are just missing the targets. It's tough to do.