If your place of employment is open, and you have the opportunity to work, then there is absolutely nothing rude or greedy to ask to work on those days. Now, that being said, the company could say no, but it certainly won't be a negative mark against you for asking. In fact, that you are clarifying the situation would be a good mark for you, I would imagine -- it certainly would be for me if I were your boss.
The situation would be different if your place of employment were not officially open, which you've carefully noted is not the case.
In my own company, we have several different types of employees -- full time salaried who are overtime-eligible, full time salaried who are not overtime-eligible, and contract workers who are also overtime eligible. We have company holidays -- days when the business is technically closed -- as well as floating company holidays -- days when parts of the business are closed but others aren't -- and we take this approach: if we are open, we work. If we are sort of open and you want to work, then work. If we are not open, but there is work to be done and you want to do it -- regardless of your status -- then say so and we'll pay you to work. For us, it comes down to communicating expectations (both sides), and I would imagine it's the same in other places, so just have that conversation with your supervisor and please don't worry you're stepping out of line, because you're not (from my perspective).