Can a university in the U.S. legally force students to get regularly tested for COVID-19?

I attend a university in the United States which has recently released its COVID-19 response plan, and part of that plan includes requiring that students get testing before coming to campus, and then get tested regularly while on campus.

It should be noted that all contracts/agreements (school policies, dorm contracts, acceptable use policies, financial aid agreements, etc.) were signed months ago, and include no reference to COVID-19.

Can the private university legally require that students get tested before returning? And if they can (say, due to some new state law or something), can they also require you to submit to regular testing while on campus? My first reaction is that this would be a simple violation of 4th. amendment rights, but because there may be exigent circumstances, are there exceptions, or have I simply misinterpreted everything?

Say, for example, that a student returned without getting tested, and the administration then proceeded to expel the student, would the student have a strong legal case against the university? A case that they would be likely to win?

A few things to note:

I’m not providing an opinion as to whether or not mass testing of students is a good or a bad thing, and I’m not looking for moral arguments in return. I am simply looking for a healthy, factual debate as to whether or not it would be legal.

I suppose that the university could impose micro-restrictions. For example, they may require you to get tested before participating in clubs or sports, as those contracts haven’t been signed yet and they could make it part of the agreement. However, in this case, I’m simply referring to just returning to and living at school.