I don't know about where you are, but here there employer gets to say when you're allowed to take vacation time, and you're required by law to give reasonable notice of resignation, which is generally understood to be two weeks.
Can you cite a statute?
Yes, you are required to have an agreement with your employer for vacation... if you're planning to come back. But if you don't show up and they fire you, or you quit, then they're obliged to pay you out for any banked (paid) vacation days which you've earned.
It depends on the state, but many states either do not have a law or only force the companies to pay out accrued vacation if they don't have a policy that specifically says they don't pay it out. Here is a list of the policies per state, here is a PDF that has the actual statutes involved for each state.
Of course, they can just do like my company has done and get rid of accrued PTO altogether. The new policy is to take time off whenever you want. Of course, this is good for the company in two ways - they don't have to pay vacation out when they lay people off so they don't have that liability on the balance sheet and people will take fewer days because they don't know how much time off they are allowed to take. The slackers who took a lot of time off with PTO will still take a lot of time off, but the people who don't normally take a lot of time off will probably take even less. Win-win(company)-lose(employee).