If they're doing damage then they gotta go,,,, it's as simple as that !!
As for the hawks snatching a squirrel,,, I saw the same site one time in my backyard as a hawk swooped down and nailed this nutter that was hanging out near the top of the tree and not paying attention... That was the last mistake that squirrel ever made. I don't have any hawks nesting within site of my house but I do have a couple that routinely buzz the air space above my yard. I guess they've noticed that there are always birds at my feeding stations and they've now included my air space to their normal patrol area. I actually have 2 different kinds of hawks patrolling my yard, one is rather large and the other is much smaller and a different color. The smaller ones favorite meal is mourning doves. It waits until there are several doves feeding on the ground and then all of a sudden the hawk dives in at incredible speed and slams into one of the feeding doves. The result is an explosion of feathers and the hawk standing on top of the dove with it's talons firmly clenched into the doves body. I've seen this bird score a "hit" several times and each time I just stand there amazed at the speed, accuracy and killing power of the bird.
Jeff