Hi Neel... Great question! I believe that the end result is the most important thing here, and that is that you're getting a degree from an accredited university. That's what employers will be most interested in.
You could list the other colleges attended on your resume, depending on how long you've been at the latest place. You don't want to lie and say you were there all 4 years, if it was actually only a semester.
If you decide to do that, I'd explain the moving around in your cover letter with a simple statement, such as, "My frequent transfers to different colleges are related to serious illnesses and relocations." I'd leave it at that and let them ask you about it at an interview, if they want to.
Alternatively, you could just list the current university and degree achieved, without any dates, on your resume.
I don't think, though, that you're going to be penalized. After all, you've shown tremendous determination in continuing on, despite your many challenges.