Does Power of Attorney give a person the right to determine who gets to visit them if they are in a Board and Care Facility?

I would contact the ombudsman for the facility and ask them what the local rules are. Your l9cal counsel on aging can provide you all the numbers and they can probably give you some information about how that works.

I believe that it has a lot to do with mental capacity. But maybe not.

I was told that I could decide who came and visited and who my dad could leave with. I declined, please somebody come deal with him for a while.

This seems to vary a lot between facilities. Some seem to take the word of a relative who claims to have a POA, even without taking details of the POA. Others respond as with DollyMe. Unfortunately, challenging it is difficult and expensive legally. In answer to OPs question, I'd ignore what you got told, make the visit and see how it pans out.

Under a health POA, and if visits are exceptionally disturbing and agitating for a patient, often visits can be limited after consultation with doctor and POA for health care.

All I can say is that, we asked that question at the home we just placed my step father and his wife in. We are concerned about her son who is a career criminal. The answer was no, he could come in, take her out, no approval required from the POA, who is my brother. They also stated that we would need to have guardianship over her, we are going to pursue this.