While watching "Starship Troopers" the other day I noticed that both men and women were taking showers together, training and fighting along side. Are the sexes equal in that world/universe? Could women become a "citizen" ? Or is the showering scene more of a director's choice?
To answer the primary question, in both the book and the movie women had 100% the same right to enter the Federal Service and through it, to become citizens.
AncientSwordRage's answer addresses the movie, so I'll address the book.
The official rules are cited by Sergeant Ho at the time of Rico's recruitment:
But if you want to serve and I can't talk you out of it, then we have to take you, because that's your constitutional right.
It says that everybody, male or female, shall have his born right to pay his service and assume full citizenship - Chapter 02
“Huh? How can he stop me?” And of course he couldn’t, not legally. It’s the first completely free choice anybody gets (and maybe his last); when a boy, or a girl, reaches his or her eighteenth birthday, he or she can volunteer and nobody else has any say in the matter. - Chapter 02
Moreover, there are many examples of women serving, including Rico's school-age romantic interest, Carmen Ibañez (a Navy junior officer when we see her later in the book), and already-serving women in the Navy, especially the Troop ship captains (e.g. Yvette Deladrier commanding "Roger Young").
On the other hand, like many things in Heinleinverse, and especially Starship Troopers version of it, things are extremely practical. In other words,
women are concentrated in the Navy, especially as pilots, due to smaller size, better tolerance for high Gs and faster reflexes.
When a female pilot handles a ship there is nothing comfortable about it; you’re going to have bruises every place you’re strapped. Yes, yes, I know they make better pilots than men do; their reactions are faster, and they can tolerate more gee. They can get in faster, get out faster, and thereby improve everybody’s chances, yours as well as theirs. - Chapter 01
Besides the obvious fact that drop & retrieval require the best pilots (i.e., female), there is very strong reason why female Naval officers are assigned to transports: It is good for trooper morale. - Chapter 13
We did go forward for guard duty, because the Rodger Young was a mixed ship, female captain and pilot officers, some female Navy ratings; forward of bulkhead thirty was ladies’ country—and two armed M.I. day and night stood guard at the one door cutting it. - Chapter 10
OTOH, in the book, there didn't seem to be any women in Mobile Infantry (while not commented on, it seems Heinlein painted MI physical requirements as too much for a woman).
But if you didn’t have more urgent things to do after supper, you could write a letter, loaf, gossip, discuss the myriad mental and moral shortcomings of sergeants and, dearest of all, talk about the female of the species (we became convinced that there were no such creatures, just mythology created by inflamed imaginations—one boy in our company claimed to have seen a girl, over at regimental headquarters; he was unanimously judged a liar and a braggart). - Chapter 04
Also, there definitely are women serving in the armerd forces outside the Navy - namely, Carmen talks to Major Rojas during recruitment, who is a female, and holds a non-Navy rank of a Major.
Just to be clear, there seemed to have been absolutely no prohibitions or limitations stated in the book referring to women being in any of the branches, as long as they qualified. The differences between MI and Navy appear to be purely practical ones.
There was also a side story clearly illustrating that the gender IQ differences weren't an issue at all in Starship Troopers universe - Captain Jorgensen of "Tours" tutored Rico in math during his officer candidate tour (and was implied to have been the most qualified person on the ship to do so - not a big surprise for a starship captain coming from pilot track). It was also implied that Carmen was very good at math, way more so than Rico :)
As a separate note, as noted above, there wasn't any hesitation about placing women in command of a starship (eat your heart out, Roddenberry).
To address your last comment about shower scene - yes, that was 100% director's choice, and contrary to the spirit of the book (surprise!). On the ship, the female officers lived separately (separate quarters) and there was even a guard post guarding said quarters, though it was kind of implied that the latter was mostly for ritualistic historical reasons rather than practical or moral ones.
Women in the Starship Trooper movie universe, definitely have the same rights to citizenship as men.
Quoting form the Movie Script
DJANA'D I'm going in for politics. You gotta be a citizen for that. So here I am. KATRINA I wanna be a mom. It's easier to get a licence if you've served SHUJUMI Federation's gonna give me a scholarship when my hitch is up. I wanna be a neurologist, study brain chemisty and associated stimulus...
These are the women who explain why they are going for citizenship. Not only does Djana'd want to be a citizen but also into politics. I would say that's equal rights.
It seems that is the whole point of citizenship, sure it splits people into citizen and non-citizen. But everyone in those tiers has the same rights.
To quote the film's screenwriter (Ed Neumeier) from the Director/Screenwriter audio commentary:
It's actually a kinda society that works pretty well. There's no sexism, there's no racism and we'll see later that there's very little crime.
In fact, they seem to have achieved the ideal politically correct society, except that we question how they've achieved it...
So there you go. This fascist utopia seems to have its good points.