The other poster who was ticketed $500 was fined for using a pellet gun to kill the vermin. He may have also been cited for "discharging a firearm within city limits". Texas law considers air guns/pellet guns as firearms (oddly they also include crossbows, bow & arrow, and anything else that launches a projectile). Air gun/pellet gun hunting was illegal in Texas until 2014. Squirrels are considered a game animal in Texas so killing them can be considered "game hunting" with an air gun if the authorities want to be tough about it.
That said, Texas law does recognize "nuisance fur-bearing animals" which includes squirrels. Under the law, you can kill animals "by any means necessary" that are attacking (or preparing to attack) your lifestock, people, agriculture...or are causing destruction of your private propertywithout a hunting license. "By any means necessary"' also indicates there are no restrictions on the size of the traps you use. Under that, you can kill squirrels that are damaging your home without a hunting license. One important point to note is that while many people feel compassion for animals, including rats and squirrels, these allowances in the law are for lethal traps. Live trapping and relocating live animals is greatly restricted and can earn you a hefty fine! In other words, you can kill nuisance animals as necessary but you cannot trap them live and relocate them. Be careful of this! You could get hammered by the city, county, AND state! That could be very expensive.
As I understand it, as of 2014, you can use an air gun/pellet gun to kill squirrelsas long as the gun fires pellets at 600 FPS or greater!
My source for this is the Texas Parks & Wildlife website. The exception is that I was not able to find the new air gun laws there. They might be there, but I didn't see them. I had learned the information about the change in air gun laws from other Texas hunters.