They ask you who your previous employer was so they can see how long you stayed there, if you moved up and to get a recommendation. Unless you left on bad terms, I would definitely list the hospital. Showing that you can keep a job for numerous years shows that you are reliable and the more job experience the better. (Even if it is not in the same field.)
The job market is tough right now. I am a college student as well, currently do not have a job and can't seem to find one. I have several years of customer service experience plus four years at a restaurant and according to Macys, Chilis and others I do not have enough experience. Macy's told me that I did not have the qualifications to work there in the shoe section when I was a personal shopper at my old job in a clothing store. Don't understand it but the more job experience the better. Another one that baffles me is that I used to work for Hallmark and have plenty experience, just re-applied and they said I do not have enough experience! I worked for the company at a different location a few years ago for over a year.
Sorry to bring up my problems in your question but I just want you to be aware of how much having experience can make a difference whether you get the job or not. As I said, with plenty of experience and all great recommendations I still cannot land a job! Your answer is no, you do not HAVE to put your last job down. I would suggest it for a better chance at a job. Best of luck to you and your job hunt!