Oddly, this fear of going suggests something mental, not medical. Not mental as in there's something wrong with her, but mental as in she's picked up the idea somewhere that going poo is a BAD thing and she's clinging to it.
I'm wondering if her caregivers were loudly dissapproving of her or another child goiong in their pants and that started it. If they are attempting to potty train her and are using more agressive verbal messages it tends to cause a child to just kind of "fear" going at all.
My oldest daughter had a similar problem. It turned out my then husband, would yell at her for either a)not wiping properly when she tried on her own or b) yell for her going in her pants when she was thought going at all meant she'd be in trouble. (And yes, I took care of the situation between them-lol. He's no longer in our lives.) Kids look at situations a lot differently than adults do. And they have a reason for everything, even if we don't think they should.
I think maybe talking to her calmly, getting her some kid level books about going poo, and maybe letting her see that other kids do, in fact, go poo all the time might help out a lot.
And don't make a fuss if there are accidents. Just calmly state, "Oh, you had an accident? Ok, let's get your pants changed then..." When you don't make a big deal out of it, some of the fear will be gone.