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.
The official rules are cited by Sergeant Ho at the time of Rico's recruitment:
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,
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
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.