Roxy - aka The Squirrel Hunter - lives with me. I exist often in her world to let her in and out of the house between hunts! She doesn't always chase them and bark, she actually lays in wait, I've even seen her point like a big old hunting dog. lol
She'll watch them quietly and then when they least expect it go after them. Her favorite activity may be treeing a squirrel and keeping them up there for as long as possible - this does often involve barking at the squirrel and definitely circling the tree.
If Bubba goes out with Roxy for a squirrel hunt, he has no patience, he just charges after the critters.
I laugh at how some of these squirrels are almost as big as little Roxy. I agree there is not a lot you can do about this - it is innate. Yorkies were developed to rid the world of rodents. They are doing what is natural - the best thing I try to do is not let her keep a squirrel trapped in a tree for too long.
Sometimes I do have to go and pick her up when there is a squirrel in her sights. How could she come in when there is a squirrel invading her property