I'm sorry to hear that this happened to both of you. I have been lucky to work for someone who has similar issues in his family so he is very understanding. What I find really ironic about all this is that you are the people that are aware of your condition and are getting help. The people that they should likely be more worried about are those who are in denial and ignore their symptoms.
IsabelAmy, as you know working in animal rescue is not easy; it isn't a high paying career and has a lot of stress associated with it, so I admire your passion for the work. The animals certainly need people like you! It smells to me like there is some ulterior motive involved and they are just using this as a convenient excuse. I've seen this happen too many times, not just to people with mental illness but for other reasons as well.
Perhaps you can go meet with people from other shelters and see how they respond - perhaps you can volunteer for a while and they may hire you once they see how great you really are? I know that's not an ideal option, but at least you'd have the opportunity to show the real you rather that what your boss may be saying about you.
I hope everything works out for you! I'm sending good thoughts your way!