Is there any problem with students seeing faculty naked in university gym?

I think that if this is a shared student/faculty gym with open showers, there’s an expectation of seeing some of your fellow university goers naked. I wouldn’t worry about it. Your students are adults and this is the norm at the gym.

You’re not overthinking, you’re just... thinking, and that’s perfectly reasonable. However, neither were the university authorities underthinking it when they decided to set up a gym that’s open to a mixed crowd of students and faculty. Nor were they underthinking things when they wrote the faculty code of conduct (or whatever the equivalent thing is called at your institution) and did not include a prohibition on faculty using a recreational facility used by students.

The reality is that there’s a false premise floating around — a pernicious belief about professors that pops up every now and then and seems hard to shake, even for people who are themselves professors — that professors are some kind of special breed of humans, or a caste of high priests that need to set themselves apart from the rest of society. We’ve seen this come up on this site in questions about professors swimming, using dating apps and doing other completely normal and mundane activities that no other person will ever think of asking for permission to go about doing.

Well, we need to be very clear about this. This belief is completely and utterly false. Professors are ordinary humans and when they are not at work they like to do the exact same activities as everyone else does. They go to the gym, they date, they swim, they eat the same food, shop for the same groceries, etc, and they have naked bodies just like everyone else does, which they sometimes expose in (gender-segregated) showers at local gyms just like everyone else does. Anybody who has a problem with their professor doing those things needs to simply get over it.

I do understand (from experience) that for a professor in a small town you don’t necessarily always want to have a chance of running into your students everywhere you go about your daily life outside of work. If you personally prefer to work out and shower at a gym where that’s not likely to happen, that’s perfectly reasonable and fine. The point is, you should choose which gym you go to based on your own personal preference and convenience, not because of second-guessing how your students might feel about seeing you, or because of feeling hamstrung by some perceived social taboo against professors mingling with the rest of society. If such a taboo ever really existed at all, it was in some long bygone era, and the perception that remains today is simply an anachronistic vestige of those days. We need to let this misguided notion die.

TL;DR: professors are people.

As a student, I would find it mildly uncomfortable to see a person in power over me naked, (especially if you have a different gender than me), more so if I didn't expect to see someone I know there (in big schools, I might expect to be completely anonymous). However, while this might be an embarrassing moment, I wouldn't (and no other serious person would) hold it against you that you use that gym. So, I would advise you to not overthink this and carry on as usual, but do not make the naked moment more awkward than necessary (i.e. don't start a conversation about research while naked).