I have enough rooms in my schools (Grade and High schools) but I have no students coming in from the other cities in the regions. I don't see where I have the option to turn that on in the regional view.
To attract students to your University, simply having a connection to the region is sufficient. Many students will borrow their parents car to drive to university and this only requires a road connection to your region. For younger students, their parents will drive them across region to school if required.
As with all inter-region traffic having a strong mass transit presence (municipal buses and trains) will provide additional options of commuting.
It's worth remembering that clicking on individual sims will tell you where they're from and where they're trying to go, and around school time on the roads near your education buildings you will often be able to determine how your students are arriving.
Do you have a municipal bus garage (one that serves the city and the region) or heavy rail links? Given that most students can't drive in real-life they'll probably be dependant on mass transit to reach your city.
ASIDE: I've also found that in general where your 'Sims' have the option to move between cities they'll do it automatically, and you'll see Police/Fire/Ambulance from other major cities even if you don't allocate them volunteer vehicles. So I don't know if there's buggy behaviour around inter-city transport of if that was the original intention