I can't speak directly to a career in health, I can only speak for my experience for my grad school experience at Santa Clara University for my Masters in Business Administration.
A marketing professor gave a really great talk one day and it went something like this:
"You guys are at SCU. You know what this means?
You have to work HARDER. You have to be BETTER.
Why? Because you have to compete with those two little schools down the street. You know, Stanford and UC Berkeley.
You know what an MBA means coming from those schools?
It means much more than an MBA coming from here. On what? Name alone.
So my job as a professor here at SCU is to prepare you, to work you and make you a better performer than those guys. Because at the end of the day, when they put an MBA from SCU versus an MBA from Stanford, they're going to pick the MBA from Stanford, one hundred percent of the time.
What will give you the edge? The fact that I'm going to put you through the wringer and make you do corporate level work and extremely critical thinking. You're going to get that interview and you're going to DEMONSTRATE that you're better than that Stanford or Berkeley kid.
That's how you're going to compete. Because you're not going to do it by comparing your pieces of paper."
LONG STORY SHORT: All things equal, your school matters. Employers will pick the better school. However, be the better candidate by having more relevant experience. Be the top candidate by having both.
Hope this helps, best of luck, get into the best school you can get into!