Even though he still gets embarrassed to cover himself up anyway (of course in the context of the trailer, it's implied someone out there was watching him anyway), being in an isolated scenario like a bathroom certainly isn't the same unless they were being jokey about it in a fourth wall manner. They do seem to suggest these animals are at least aware of clothing as a social contract and perhaps got it ingrained in their minds over what is considered decent or not, at least through proper condition as we see the animals in his civilized world do. The ones that don't (like the Mystic Springs patrons) obviously explore it as a temporary lifestyle.The notion of Nick not having anything down there to be ashamed of can be correct, but the whole nudity issue and needing to cover the area is still a very human gesture we tend to take for granted, even with cartoon animals, and we've seen this played out many times before. Whether there is something 'important' or not doesn't seem to matter as long as one feels concerned for appearing indecent to others.