Lots of experience being the person who's being steamed at here, and I always use the same approach. Doesn't matter how stressed or crying or yelling the person is, I always go with the following:
Hold both hands out, palms towards the stressee, and in a firm voice say "Wait." When they give you their attention, do a rolling motion with your hands and say "ok, first, breathe", and take a large breath. The stressee will unconsciously do this with you. Then, holding your hands palm down, say "now, calm breaths." Make the rolling motion again and breathe with the other person twice. Then offer a tissue box if available, or some water if there's any to hand, while saying "now, steady and calm." The tissues allow the stressee to focus on something else and breaks the moment.
Now you have control of the situation again, gently but firmly state what will happen. "Ok, you have an appointment with Professor Dinglebat at 3pm. We have a small cafe you can sit in and gather your thoughts until then." After that, you need to play it by ear.
The important bit is that you take control of the situation, reiterate what is going to happen, and be firm.