Well we'd need to establish some rules, before anything else. First off, what counts as a holiday? Obviously I think religious ones like the winter holidays should count, but what about the "Hallmark holidays" like Mother's Day and Father's Day? What about government-recognized holidays? What about social awareness holidays like Earth Day? Should famous birthdays count? If so, where is the line drawn on how famous that person needs to be? Jesus' birthday is kind of a big one, but how about Elvis' birthday?
I think the best way to figure this out would be to do research on world holidays and start putting together a calendar. I can see a really interesting and informative video about this.
January 1 is special for being new year's day. And if any days after January 1 are not special, then the earliest of them would be the first non-special day of the year, which is special in it's own right, giving a contradiction. Therefore, there are no non-special days. They are all special, so make the most of each one.
You have to remember, some religious holidays follow calendars other than the normal one, e.g. Easter follows the lunar calendar, so changes each year, so what might be a special day this year, isn't next year.
Other days with no national holidays in 2024 are:
7th March,
3rd April,
11th May,
8th June,
23rd August,
22nd October,
13th November,
27th November,
14th December,
28th December
Hmm, I have a calendar somewhere that includes holidays and events from around the world. There are actually quite a few days with no events or holidays, so it's not unheard of. My guess is that a good candidate is likely somewhat distanced from seasonal constants like the equinox.