OK, I have a different perspective. 20+ years in a "traditionally male" field. I see two unrelated issues in your letter. The first is whether or not you need a witness. Only you can answer that, given the situation. A manager with any sense should know that if an employee wants a witness, they should have one.

Second, you can't cry. You just can't, not over work stuff. To avoid it, rehearse mentally beforehand. Tell yourself regarding the boss 'I don't care, you can't hurt me, your opinion doesn't matter, you are too dumb to get emotional about.' Think about the subject, when you get to the point in the exercise where you are going to cry, find something else to think about to stay on topic. Take a deep breath, or two, then keep going.

If you are sad over personal issues you don't have to be in such tight control. But if it just job stuff, don't give them the satisfaction of seeing they hurt you. Part of being in the workplace is appearing tough and in control.