This is exactly what my German Shepherd does all day at the back door.
I throw out peanuts a few times a day, the squirrels come, she pounds on the door with her paws at them, and I come let her out to chase them.
They always get away.
But two days ago one of them took a similar direction as she did, and the squirrel refused to break off. it ended up about 2 inches away form her running parallel with her.
All my dog had to do was reach down and bite, and she would have had it, but she did not.
I now think she just wants to herd them.
We had a Golden/Brittany mix years ago that loved to chase squirrels and, like yours, never seemed to catch them. Except, one time, boom, the squirrel was DRT.
After that, the dog always seemed to give the squirrels a head start in the chase, and she never, ever, caught another one. Unlike our current setter mix whose greatest delight is sneaking dead baby bunnies into the house (sigh).
They just like the thrill of the chase.