Dogs bred to do specific things tend to be naturally better at those things. Feists have been used to hunt squirrels for a long long time and are very good at it. If your goal is to "keep your yard free of squirrels" then getting a dog bred for chasing squirrels makes the most sense to me. But yes, they are not trained to catch and kill squirrels. You train them to tree squirrels so you can shoot them. However, a dog that is bred to chase squirrels shouldn't be too hard to encourage to kill them. Play fetch with a dead squirrel and reward the hell out of the dog when he brings the dead squirrel back.
And I don't think a dog needs to kill the squirrels to keep your yard free of squirrels. The pressure of danger alone should keep them at bay.