Most Americans would agree that they certainly do. But when you ask the same people whether businesses have the right to not hire whomever they don't want for a particular job, most of them will say that it depends on the reason someone is not hired. And the same thing is true if you ask the typical American whether businesses have the right to fire whomever they want from a particular job. Most of them will say that it depends on the reason someone is fired.
If you really want to complicate matters when it comes to hiring and firing, just introduce the subject of religion. What role, if any, can or should religion have in the hiring and firing process? Obviously, if someone is interviewing for a position as a minister of some sort, then everyone expects, and everyone accepts, that a Catholic church will hire only a Catholic, a Unitarian church will hire only a Unitarian, a Jewish synagogue will hire only a Jew, an Islamic mosque will hire only a Muslim, and an evangelical church will hire only an evangelical.