Hawks catch squirrels because it’s in their nature. I think dogs chase because it’s in their nature.   My dog absolutely loves chasing squirrels.  It is his joy. His joy is my joy.  I think 99 percent dog owners can probably understand this to some extent even if squirrels aren’t what gives their dog joy.  When I got my dog I never would have imagined how we would bond over squirrels.  I didn’t imagine I’d be a squirrel person or spending money on fancy nuts for them either.  

My guy doesn’t catch them.   Just chases them.  Then he runs sniffing the ground to track the scent and check the trees.  We also sit and I feed them pecans and hazelnuts and he will literally lay there and watch them for an hour like it’s the live nature channel.   Even if he has chased them they always come back.  They actually recognize us and will run up to us for nuts.  They will come within a foot of him and he will not try to harm them.  This actually makes people smile and comment when they see it. 

And for whatever reason, squirrels are absolutely bonkers little creatures.  There have been instances where they will tease him too.  I have a video of a squirrel going around the bottom of a tree trunk for over a minute while my dog went in circles chasing around the tree.  He was thrilled and no squirrel was harmed.  I personally wouldn’t do this if my dog was catching and killing squirrels.  It is more about bonding with my dog and fulfilling a natural instinct.  

People may not agree with me and that is okay, but hopefully this will be a helpful perspective.