The next RNG day should be selected deterministically using a random number generator.

That's how it works, next year it's February 28th.

To appropriately celebrate this, shouldn't there also be a chance that it could occur twice in a year? And to that point, also a chance it won't happen at all?

I think it's a tricky thing to define "appropriateness" in this context. A sufficiently complete random generator could return a new universe. Which is to say, ya gotta draw the line somewhere.