I have a couple of big silver maples, 40 years old and about three feet
in diameter.

stuff snipped

What is the best way to stop the damage and make the trees live longer?

I'm getting ready to fabricate traps out of 55 gal barrels, and some 4" PVC.
The three foot or so PVC has a tipping point slightly off center so it
resets itself. It is open on both ends, but placed so the squirrel goes in
one end. There's a trigger on the bait. When he reaches the bait and
trigger, he's well past the balance point. When he hits the bait trigger,
the tube is released, and it pivots. His weight causes him instantly to
plummet into 18" of water. The tube resets itself under its own weight and
balance point. The bait stays put, either being peanut butter slathered
inside the tube, or pecans wired on the trigger. It's a reusable trap and
resets itself. I built one last year, and it worked good. I intend to make
four this year so we can try to keep the squirrels under control and out of
our fruit trees.

Steve