At the moment I am in a hotel and a gentleman asked me if I am going to need housekeeping today. My answer was: "No, thank you. I don't need housekeeping today". Was that correct? Should I have said something like: "I won't need housekeeping today" or "I won't be needing housekeeping today"? I think I may be overthinking it.
Technically, the differences among the variations you shared are a matter of tense. "Don't" is a contraction of "do not," so if you want to be pedantic/strict about your grammar it would make sense if you meant, "I don't need housekeeping RIGHT NOW," or simply said, "I don't need housekeeping." However, what you are stating is that you WILL NOT need housekeeping services for the next 24 hours--it isn't strictly a present condition. So "won't (will not)" might, by a small margin, be the more nuanced choice.
However, even clearer than the future simple in this case would be the future progressive, which speaks of a continuous action over some future period. You will continuously not be needing housekeeping for at least 24 hours, so a pedant might endorse your last option, "I won't be needing housekeeping today."
The actual truth, however, is that "today" is a very flexible word that can be used in most past, present, and future tenses. No one would quibble with any of the options you presented and they all "work." The only case in which I could see any of them causing confusion would be if the person you were speaking with was only just beginning to learn English as a second language and might only be familiar with some forms of the future tense--in which case your choice is almost certainly the simplest even if it isn't the most nuanced.