There are no federal or state laws mandating paid vacations or holidays. That's entirely at the discretion of the company.
There is no federal paid sick leave mandate. Some states have them.
We have a federal right to 12 weeks of unpaid leave for qualified medical and family situations under a law called FMLA, the Family and Medical Leave Act. Even that has limitations - small businesses, highly compensated employees, and new employees are carved out.
When you are hired, all of these are defined by the companies policies or what you can negotiate. If you have paid vacation time that accrues it'll be paid out if you quit; most companies don't like that, so they give you "unlimited time" on the approval of management. But because we have no statutory protection, your unlimited time could legally be "zero hours".
Or, you get states like here in Texas, where the state senate just passed a bill making it impossible for cities like Austin to require companies provide sick leave.
Where I am in Canada (and I imagine many states) there are legal minimum requirements so I officially get "3 weeks" vacation, but take a couple weeks on top of that using the "unlimited" policy. So "unused" vacation would be paid out according to the official 3 weeks policy in my case. I could see something similar being possible in Europe, but I think in practice europeans already take more paid time off than most North American's with unlimited vacation.
Years ago, the company I work for switched from accrued to unlimited PTO at the end of the fiscal year in order to make the books look better. When it was announced, they basically told everyone to take the remainder of the month off. The fiscal year happened to be the same as the calendar year, so most people were taking huge chunks of time off due to the holidays anyway and didn't really care. For those who had more PTO than what was left, they were told to work with their managers to arrange for as much time off as they felt was necessary in compensation.
This only worked because it was a one-time deal and the managers were (and mostly still are) very reasonable and supportive people. A while later we were merged with another company and they switched us back to regular PTO to be in line with their own HT policies.