I'm thinking that Facebook may not be the best venue for something like this. And I can sort of see the pastor's reasoning -- it *is* perceived as a "public space," and even though this particular group was privacy-protected, it may not be clear to someone inexperienced with FB what is private and what isn't. I know I can't always tell.
It's kind of like posting something on the wall (actual brick-and-mortar wall, not FB wall!) in the ladies room or an employees-only area at work. The intent may be to keep men or non-employees from getting access to the info, but the fact that it's out there for all who pass by to read sends the signal, "This is not top-secret confidential information." So there's nothing to keep those "authorized" to read it from passing the info around.
My suggestion would be to scrap the FB group in favor of updating folks on sister's condition by e-mail (it's pretty easy to set up a group mailing list in most mail programs), or keep the FB group but post some sort of disclaimer on its FB home page that says something like "All information shared within this group is confidential and should not be shared with anyone outside the group without the express permission of Animala. If you're not sure who is and isn't in the group, check the member list."
(If you were to go the e-mail route, it would probably be smart to include a "don't share without permission" disclaimer in the sig line as well.)