Well, talk about getting overly serious about a non-serious issue. As with other firearms, there are several factors, one of which is definite shot placement. In another thread, the NAA mini revolvers are being debated as "better than nothing" firearms to carry. The point here is that with proper shot placement, the pistol you have pictured or a Daisy single pump old fashioned BB gun could take out a squirrel, but unless shot placement is right on, probably not.
Your best bet is to go with a BB rifle, whether you shoot BBs or .177 pellets. As with other firearms, penetration is the key. Plus with the rifle, you can do a lot better with shot accuracy. Growing up, I had a Daisy Powerline 880 and my brother had the 881 with a rifled barrel. He always shot better than me, but he put a scope on his and could hit just about anything he wanted, where he wanted, out to 30 yards when firing fully pumped rounds. I just had iron sights and no rifling. I could, however, hit a squirrel at that disance. My brother could pick what part of the squirrel he wanted to hit.
With the pistol, you won't get the long axis aiming precision as you would with a rifle, even if the pistol is suitably accurate for the task. The best shots are going to be head shots, followed by shots to the upper chest cavity, much like you would shoot a deer. You need to have sufficient power to penetrate fur, hide, and some bone structure including the shoulderblade and ribs on the torso. A head shot will not encounter as much bone as the shoulder/torso shot. Squirrels don't have very massive skull structures.
Look at it this way, the squirrels will continue to do damage. That initial $30 cost you were out for damage will be matched again. Spend a little money on a good BB/.177 pellet rifle that can be pumped to high pressure and it will pay for itself in short order as long as you don't end up shooting the neighbor's windows or pets. Plus, they are a lot of fun to shoot.