You were going to leave so they could have a session about your boyfriend's depression. What I think I would do is have your boyfriend take care of his problems, get his own therapist that takes his insurance, etc. and not get in a confused mess about what is happening with any given therapy session.

You spoke to the therapist about your problem, feeling you had done something bad to your boyfriend and she reassured you and recommended some changes you found helpful, etc. This appointment was for your boyfriend and his problems; they cannot overlap with yours because you are a different person! If you were going to leave so they could talk, I would have gotten up to leave, especially after I started to cry about my issues?

I would go see the therapist alone again and work on clarifying what is your stuff and what is someone else's. Your boyfriend's feelings are not about you; he can feel happy, sad, angry, whatever he wants/feels but they are coming from inside him and are not about you and your issues just like your crying is about your issues and not your boyfriend.