Does the college you go to determine the places that will hire you?

It depends on what profession you are considering once you graduate. If you are looking at medicine, law, or another high level profession - your alma mater might add prestige to your resume - assuming you did well in your courses, were involved in campus activities, and can present yourself well in an interview. Some schools may not have an Ivy League name, but are known for certain programs. If you are interested in a athletic training, environmental sciences, or anthropology, for example, you may want to look into schools that have strong programs in these fields.
More often than not work experience and accomplishments will begin to trump your college selection as you move deeper into the workforce.
Nursing is offered at many institutions so I would look into a program that meets your needs and expectations as a student from class size to campus feel. If you are happy where you are enrolled, it will make all the difference academically - no matter the name of the school.