A while ago, I had this come up for myself, so I hightailed it to my BON website. There, I learned that I am NOT required to work overtime, that once my shift is over that I was entitled to leave provided I reported off to an appropriate provider (in my case, it was the nursing supervisor). My BON did NOT consider this to be patient abandonment, but rather an employment issue. And this BON did not interfere with employment issues.
(Mind you, this is just 1 BON of 51, so your mileage may vary).
Now, my job could have very well decided to sack me for refusing to stay over--I was in a Right to Work state and in a non-union facility, so I didn't really have a lot of protection had they decided to go that route. But if they had, the BON would not have interfered because it was...you guessed it, an employment issue.
Check with YOUR BON to see their stand on forced/mandatory OT. Then weigh out the risks/benefits of turning down overtime, especially if you don't have a union or P&P in place to protect you. Maybe take it once in a while to show you're carrying your share of the burden...but do not feel obligated in any way to volunteer for every shift.
And to turn it down, all you need to say is, "sorry, not available that day."