Hunting the squirrel is an a slang term for the late 1700s and early 1800s to refer to a particular type of hooliganism.
As the use of carriages became an important part of running a large household or company (including the ever-present post office), more and more lower-class working stiffs were employed in jobs that put them in the driving seat of large four-in-hands, stagecoaches, and other carriages designed to be both large and speedy. At the same time, young, wealthy wastrels were also known to pride themselves in being able to handle a four-in-hand, and it sometimes happened that one of them would make a pest of themselves on the road. Given these various conditions, it was not uncommon for road rage to rear its ugly head, and one form it could take was 'hunting the squirrel'.
This was the act of coming up alongside a smaller and lighter carriage, such as a chaise, and harassing it -- crowding it, pulling up in front and blocking its path, bumping it, and perhaps even forcing it off the road. Reportedly this was often done to women and other defenseless or inexperienced drivers. To be fair, the 'squirrel' might often be a young fop in a racy phaeton or curricle; these types were known to drive with gay abandon, cutting off other drivers and spooking their horses.