When a schedule has already been posted up for two weeks and the employee wants the day off. Can you tell them to either find a replacement or come to work? Is that in legal bounds.
California has "At-Will" employment. Wanting to not work on a certain day does not make you a member of a protected class. As you did not mention why you want that day off, I guess it's not a protected leave of absence either (like jury duty, voting, military service, caring for sick relatives, pregnancy etc).
So if someone just wants a specific day off... he or she can certainly ask. But if the employer says no, then tough luck. You can be fired for not showing on that day or you can be fired for no specific reason at all. And the employer is free to change that as they see fit.
It depends on what you mean by "require". You can't take a gun and force the employee to come to work. You can tell them that they either come to work (or find a replacement) or find a new job.
People have the right to take holidays, and you have the right to refuse the holiday if there are business reasons. And then employer and employee should behave in a reasonable way.
It sounds like the employee wants the day off at very short notice, too short for you to adapt to it. In that case "you can't have that day off unless you find a replacement" is reasonable. You could check if there is a very good reason (kid in hospital is a good reason).
If you don't allow the holiday, and they don't come to work, then you can look at whether you want to employ them, or take it into account when you do a performance review to determine raises, and so on.