A few days ago, leaving the Y and half delirious (as I always leave that place), I noticed a few twigs. The next day the sidewalk was covered and this time the tree was an old maple - I think red (Acer rubra).
I talked to our resident squirrel expert, Dr. Mike Steele, and it ends up this is a well known but not well documented behavior. According to Mike, squirrels bit the ends of the twigs off to get the sap that drips from the end but this also serves to increase the size of the samaras (winged seeds) that come from that tree. He did an experiment years go to test this and the results supported that hypothesis. That study remains unpublished but should be repeated with a few more trees.
It can't just be the sap, for that they could just chew the end. But they chew both ends and drop the twig. Intriguing.