Man, saying this I am gonna admit my "bird dog " is a squireel dog but oh well, she aint a good bird dog.
I have a GSP, she hunts squirrels and trees them pretty good ever since I shot one and she got to mouth it and shake it when it fell out of the tree. Now she hunts rabbits ever since I started bringing her with my beagle. If I shoot it and she gets a taste of it she will start hunting that particular animal for the day or until I shoot something else.
My beagle has also shown an interest in squirrels but I have not shot one for him to get so he doesnt pay them that much mind anymore.
I think anydog will chase squirrels so long as you let them get ahold of one and praise them when they do.
anyway here is a link to info.
squirrel dog cetral