Someone suggested the R9. I have bought and sold 3 different price point PCP rifles. But the R9, I will hang on to for probably the rest of my life. It is break-barrel perfection and is 10 times the rifle as any PCP for the same price point.
“But a PCP at the same price point is going to be a more powerful gun, and probably a repeater, probably very quiet, etc.” Build quality. No one is sending an R9 back for a trigger that snapped off, because there is no pot metal in a R9. It’s rock solid. And my R9 shoots tighter groups than all three of those PCP’s I used to have. They were a Gamo Urban, BSA Buccaneer, and Benjamin Discovery. But, there is a learning curve to shooting a spring gun that well.
The old Sheridan will for sure lay a smackdown on some squirrels. But it’s slow to reload and slow in velocity. 20 maybe 30 yards.
As for a shotgun, I have killed a pile of squirrels with a 12 gauge and there is zero wasted meat. Up close, edge them or wait for him to stick his head around the tree to look and take it off. At distance, no issues. Tiny pieces of shot don’t really damage the meat. But if you center the pattern on him, it will obliterate the squirrel at close range. Just don’t center the pattern unless you got some standoff distance for it to open up.
This is not to discourage you from a PCP, I think that you should probably get a price point PCP to start. At least for your purposes. It probably suits you better, and you can and likely will get more pellet guns down the road.
Looks like you’re gonna be going through an FFL regardless, might as well get something with thump behind it. I recently found a great deal on a Stoeger XM1 S4. That’s a different gun from the Scout, the Scout is a nicer gun I think. But I’ll let you know how she shoots.
Looks like Pyramyd Air is offering a free buddy bottle with .25 cal Avenger in a laminate stock. Might check that out. It will wreck a squirrel for sure
https://www.pyramydair.com/product/...-pcp-air-rifle-red-blue-laminate-stock?m=5815View attachment 538615 
