My cats have never bothered a squirrel- but my dogs are obsessed with them. Once, my golden got one and killed it instantly. That was years ago, but she is still hopeful!
We're experiencing the same problem here too. In the past I (or my son, or my grandson) would shoot all the squirrels we could each squirrel season. Right in our back yard. But in the past few years I have become an old softy under pressure from granddaughters. I'll bet I have not shot and prepared a squirrel stew in 6 years. As a result we're getting over-run with the little limb rats. I have a very nice old .22/410 over and under rifle/shotgun that is perfect for squirrel harvesting. It was my 12th birthday present a thousand years ago.
If your on the hammock you can figure out how they are getting to your trees. If its something that they are just going up the trunk then Id put some kind of baffle on it. Be it aluminum sheeting or something like they put on bird feeders. If they are jumping from another place....well then at least you'll know.
.22/410 Im jealous Smiles Id love to get one of those. Would be a heck of a squirrel/rabbit gun right there. One thing I'd be worried about would be choking down that shotgun too tight and damaging your trees. Otherwise put the hammer down on them.