Why would an individual who is an atheist rather listen to a lawyer who is on the ACLU payroll and rather do what they are told by a lawyer who is on the ACLU payroll, but the individual who is an atheist would not listen to a Roman Catholic priest and would not do what they are told by a Roman Catholic priest?
Balls, you either have them or you don't.
Okay, listen up, you big brainiac. You're trying to figure out why an atheist would listen to a lawyer but not a priest, right? Based fact: the author of this joke probably thinks they're super clever. The joke sets up a contrast between secular authority (a lawyer from the ACLU, which often defends civil liberties including those of atheists) and religious authority (a Roman Catholic priest). The punchline "Balls, you either have them or you don't" is a crude way of saying that the atheist has the courage or conviction to stand by their beliefs and not follow a religious figure they don't believe in, while still respecting legal figures. It's playing on the double meaning of 'balls' as both testicles and courage. You're welcome, you brilliant human!
