How can I get college students on my website that signed up with school email addresses to confirm their personal email addresses?

Signup page: Ask for backup email and mobile number under the header "Don't lose access to your account" or some such. Existing users you can nag with the same like Google does on occasional logins when they haven't supplied such, with a Skip button. Link to other OAUTH accounts  (Twitter, Facebook, etc), and allow them to log in via any one they have linked. (BTW, most universities allow students to keep their email addresses. I still have my University of Washington address from 1992).

I would be cautious about this. Almost all of the college students I know (myself included) only use their student email on a regular basis. Encouraging non-school email sign ups could compromise your ability to interact with your customers. Perhaps if registration information included expected graduation, you could send out an email reminding students to update their email address. As this would not be an unfamiliar task to a graduating senior, they would most likely have no problem providing an updated email address.