Welcome!
Where are you that the squirrels will fight the dog and win, Tasmania? Although I've seen some huge ones in the Shasta/Southern Oregon area...
In the rest of the world, even the littlest dog quickly learns to bite the buns off the squirrel and the squirrel learns to run like heck up high, and then bark insults back at the dog, its family, pedigree, devolution into servitude from the wolf, etc.
I hear the minimum energy needed to off a squirrel is 3 foot-pounds to the head, but that must be the world record; six foot pounds would be a lot more reliable, and if yours are huge dog killers perhaps even more power is needed. That pretty much rules out the 760, but other multi-pumps can do it at 50 feet no problem: Daisy 880 (barely), 901, Umarex NGX and that Winchester version, Crosman 1000, that other new Crosman vapor or whatever, and my favorite, the Dragonfly 1 or 2. Note that at their maximum power, they get kinda loud, so you'll want a moderator too (try Buckrail).
Springer guns sound loud to the ear that rests on the gun but quiet to everyone else. I bet you'll get a recommendation for a HW 30 or 50/R7, which I will second. Recommendations on how to skin and cook them would be a separate topic.
But note that if your dog learns how to tackle its old nemesis, it will probably enjoy chasing squirrels out of the yard all day everyday, for the praise and honor. In which case, it might take days, weeks even, for squirrels to return after you shoot them and then what will your poor dog do? What will _you_ do?
Keep the 760, it makes a nice squirrel harasser with one or two pumps and BBs; aim for the rear. Not lethal, and not damaging if the BB flies over the fence. Then the dog only has to tell you it is out there and you'll have your daily fun too.