ubiquitin – I agree.
But I believe having same gender personal as a bridge is a really minor detail when someone really needs help. they would probably be more focused on getting a better facility, better doctors and people that are able to navigate the health care system and get things done. As I mentioned the higher the acuity is, the less people tend to focus on some details (such as gender) that would otherwise seem significant to another individual, this is probably true for male and female patients.
your question has a simple answer, I would want whoever I believe can better help me at the time of need. I have very rarely seen anyone being uncomfortable with a provider that is of opposite gender in time of an emergency. If this were not an emergency some might take minor details such as gender of the provider into consideration.