Interestingly enough, I'm a licensed Falconer and this is the only viable option we've considered, but it'd be a big Public Relations problem with the students/the town. The squirrels (I'm 100% not joking) have their own instagram page with 14k+followers, so me calling in some favors with some friends and hawking through campus over Christmas break would not be well received...but it's the most 'natural' form of population control we can come up with.
Encouraging birds of prey isn't as easy as you'd think...hawks are very territorial, and the ones on campus are very fat and happy, so overcrowding hawks (who don't regularly hunt squirrels, as they're far more dangerous to the hawk than mice and smaller rodents) would actually hurt their own populations, and isn't really doable.
Allowing local Falconers to come in and hunt as a special population mitigation event would, I'm sure, be effective...but getting permission from the top down might be impossible.