I like lever guns (and have owned the Browning, Winchester, and Marlin), but if I were buying a rifle specifically for squirrel hunting it would NOT be a lever gun.
For many years I hunted grey squirrels in southern hardwood bottomlands with a Remington 541-S with a Weaver V12 4-12 power scope. I still have that rifle and it is very accurate, but my current premium hunting .22 is my Kimber 82 Classic with a Leupold Vari-X III 6.5-20 EFR scope. The gun shoots under 1/2" at 50 yards with target-grade ammo, and at the range I can SEE the bullets through the scope as they fly downrange and land on top of the crosshairs. For hunting I leave the scope on its lowest power setting most of the time, but it's nice to be able to turn the power up to 10x or 12x and center a grey's eyeball at 50 yards... and then watch it disappear as the bullet smacks into it.
Although they shoot well, I'm personally not a fan of tricked-out 10/22s. Even with the best triggers on the Rugers, I find the Kimber and other premium .22 bolt actions to have a better trigger pull. I also think that the knowledge that you cannot make up for poor shooting with firepower to be a calming influence on most hunters... it forces us to slow down, aim a little more carefully, and make the shot count.
My recommendation is to look for a quality bolt-action (Kimber, Anshutz, etc.), spend the extra money, and have a precision tool that will hold its value, last a lifetime, and provide you with lots of fun and NO excuses... if you miss with one of these rifles you have only yourself to blame.
Good luck and good hunting regardless of which way you go.