Apparently it's called "latrophobia", PSTN, and I've had it all my life too. Aside from severe generalized anxiety/panic disorder, I have two specific phobias, one of which I've talked about around here and got wonderful support.
The other is absolutely anything even remotely medical, to the point I refuse to watch shows or movies with it, no matter how good they're supposed to be, and will drive out of my way to avoid passing a hospital.
Having the constitution of an army of battle mechs meant this worked great for a really long time. More recently *infer all kinds of cussing here* I burst into tears for about an hour over getting checked in to one of those hideously terrifying places. Maybe I wasn't in the greatest mood to start with, but that was about getting checked in after hanging around an ER for awhile.
Orange is right, most professionals respond very well to knowing how you feel about these things. Not all, but most, and as hard as it can be, it's always worth finding the good ones, who are genuinely respectful and supportive. I have a two hour drive to get to a primary care I'm okay with, and that hospital was in the next town; it's rated the 5th best in my country, while everyone here knows to avoid the local one. Don't settle or compromise, is my point, I could talk for a year about all I went through to find medical I can trust, but it's your health and their job to help. I know it can seem impossible, but remembering what really matters can make all the difference too.