Re: Oh the irony
they requested I sign a paper saying anything I developed belong to them, including any research I might perform.
This is fairly standard. At a uni, they are providing a lot of facilities for you. When I went, they allowed use for your own projects too.
If you have a good idea, it is often possible for you to talk to them and negotiate an exception, although they may expect something in return. You are paying them to teach you, not to help you found a new company, provide R&D equipment and resources etc.