They're only paid overtime if they actually work overtime. Now you may be right about them working overtime for a few days to try and close the park down, since it does take alot of work to close down a park, especially one like Grand Canyon where there are thousands of visitors there at one time.
But from my experience theres going to be a whole lot of "non-essential" personnel who are going to continue to work even though they may or may not get paid for it, I know my friend will not be able to leave the canyon in the event of a shutdown, and will help LE's close the park even though he is "non-essential".
I just do not like people perpetuating myths like public servants are leeching off the system or that they're lazy, or that they get overtime pay for working regular hours, or that they make outrageous wages, when it just isn't true. Most public servants are doing what they do because they love it or they want to make a real difference, the NPS is a great example, as are teachers, as are most public servants. Obviously there are exceptions to this, but I hate when people make generalizations and attack hard working people. I haven't met a frontline ranger yet who doesn't give 110% and care deeply about what they are doing. If they went in to the NPS for the money they made a wrong choice.
"Who are you guys??!!" -Farnsworth