Can a college or university punish you for things you say or do on Instagram?
They can try to do so. Are they within their rights in doing so? Possibly not. It depends on what's said or posted on Instagram. The student does have a certain obligation to live up to the school's code of conduct or other policies that may be implicated. If it say racial, sexual, if it's threatening in nature, there's going to be a certain consideration.
If it's say, simply a matter of somebody disagreeing with what's posted or said, a person has a different opinion or say if it's a matter of a political nature or social nature and somebody just takes exception to that. The school may be outside its bounds in trying to discipline a student in that regard but schools can be overzealous and trying to do so.
Having an experienced attorney advisor can help you best understand what your rights are and can help you best understand and navigate the process. If you're facing some kind of allegation involving your use of Instagram.