I don't have to read anything into it, I was raised in part by people who spoke that dialect. I can speak it. I can creep out people my age by using that dialect in its (creepy) nuances. I've got the grey hair to prove it and everything.

As it is used in that film, straight up, I assure you, it reflects sexism. And "boys" referring to soldiers isn't symmetric. How can I convey it... hmm...

"Men" as used here are tough, full adults. "This man does ..." means we're talking about someone plausibly a head of household. He's an individual. He's a complete adult.

"Girl" as used here means we are talking about females who aren't quite kids but neither are full adults. By default a "girl" is young and unmarried and in some ways fragile. She might be older and married but its a little sad she's still working then and, anyway, at most she's a mother hen to the girls who are younger and more typical of the group.

A fully (recognized as) adult woman in the workplace might be a "lady" or a "woman" depending on role and context. She could be "fine woman" and you oughta settle down and show some respect if you want to get anywhere in this organization. If youre gonna act like that why don't ya just head downstairs to the chat up the girls in the secretarial pool. Learn your place.

The jocular "the boys in this unit take a smoke break while the tanks get cleaned and readied for the next advance..." is a sign of camaraderie. It doesn't (like "girls") signal that they are juvenile, per se, it signals that they're an anti-hierarchical tough team. "girls" back then could be stretched to that same connotation but usually wasn't, especially in a non-wartime workplace.