I don’t know what you’ve heard about trying to stay in the US, but that wont really be your decision. The INS (Immigration and Naturalization Service) makes and controls that decision.
As for work, if you are coming to the US for your education, you get a “student” visa. You will be very limited in what kind of hours and on-campus job you will be allowed to work on your student visa. Usually, the jobs will cover your day-to-day needs (like coffee, toothpaste, hair products it wont and can’t be a significant amount).
I don’t think it will be enough to cover your health insurance fees which can run anywhere from $2000 to $6000 dollars per year, and are not covered by scholarships. Each college has their own plan. You are required to have US health insurance while studying in the US or you wont be able to register at the college.