Patricia, I hope this helps. You should be able to access links that will help you with the overnight camping with your horse.
Do the following;
www.nps.gov
Select Utah, select Bryce Canyon, on the left hand side of the page-select Plan your visit, select things to know before you come, select Private Horse riding information
Looks like they don't allow overnight camping in that area with your horse, but you should be able to trailer in and ride for the day. It looks wonderful! It would be quite a haul for me from northwestern Iowa, but it looks great! Hope this helps, Barbara