In Mesa it is Banner Health Services, but Mesa is mostly a bedroom community for Phoenix, where the largest employer is the State of Arizona.

Does the government count?

I've no idea.
If it weren't for the truckstops at each end of town I doubt this would be more than a gas station town on old route 66 / interstate 40. There ain't much here.
I'd guess it's the pinto beans farms, but who knows.

University
So big that happy / sad depends on the issue. It is not all of one thing.
total of 3500 employees in a town of 20,000

There are two large hospitals/health care organizations in my area. They employ a lot of people. But there is no one large employer that dominates the area. There used to be (General Motors), but they have dramatically declined. I work for a university and am pretty happy there. It has its problems, like any employer does, but I don't feel that I have to conform to any particular set of values to succeed. And NO ONE at my employer talks about religion; at least not to me, anyway.