Nothing inherently wrong with .177 pellet guns. There are some that have interchangeable .177 and .22 barrels. Same power but different presentations. Ive been told that difference is that the power being the same the .177 is better at close range where the supersonic velocity can cause shock. Particularly if you use hollow point pellets. The .22 pellets are heavier and slower. They carry farther but also drop more.

Interesting is that .177 pellets from a magnum power rifle, at better than 1000 fps, void the warranty on many commercial targets. The smaller high velocity pellets concentrate the energy so much that even with soft lead pellets the steel striker plates get pitted. A bit of steel gets vaporized in the impact. The lower velocity and wider spread of .22 pellets are not so destructive.

It also has to be noted that pellets guns are something of a risk for lead exposure. Particularly if used indoors. The high velocity impacts literally vaporized some of the lead. If you breath or eat the very fine powder created when the metal cools you, or your kids, can suffer. Smaller animals, and humans, are more vulnerable. Watch the air flow if you shoot indoors. Wash your hands after shooting or handling lead pellets. Take precautions so kids and pets don't get exposed to the lead dust. IMHO it is not any reason run away or not to shoot. Just be aware of the issue and take common sense precautions.

On light game, birds and squirrels, which don't have much meat to absorb the energy, you might try hollow point pellets. Less penetration and more impact. Pointed hunting pellets designed for penetration can travel right through light game. Potentially leaving them wounded to die slowly. I've also found that shooting squirrels it is often easier to go for a body shot. Below the arms and aimed at heart and lungs. The sweet spot is a slightly larger than a head shot so a humane kill is more likely. Tear a hole in the heart and/or lung/s and they drop. A cross-body shot with a hollow point makes the end quick.

If the gun you shoot is designed for pellets don't shoot BBs, even if the maker says you can. If you look down the bore, of an unloaded gun of course, and see rifling your better off sticking to lead pellets. BB guns don't have rifling. BBs are so hard that they flatten the rifling. Once the rifling is damaged pellets won't spin up properly and accuracy will suffer.