Definitely do not offer me a hug (or expect one from me, if you’re the one who gets teary). I do not want to be patted or cuddled or stroked unless we already have a relationship in which touching and hugging play a natural part, probably meaning that you are a family member or close friend.
Even more emphatically, do not try to jolly me or make a joke. Keep it respectful, calm, and a little bit detached, and give me a moment or two to settle down.
If you get weepy while telling me your business or personal troubles in the office, or while I’m firing you, or whatever, I’ll remain calm, hand you a box of tissues if one is nearby, and wait for you to get hold of yourself. I probably won’t look away as if you ought to be ashamed, but I won’t stare at you either. I’ll just give you some time. Very likely I’ll offer you a glass of water and then go get it for you. If you’ve said anything that seems to call for a reply, I’ll answer you in a gentle voice but otherwise say nothing else. If it goes on too long, I’ll probably ask if you’d like some time to yourself.
I don’t believe I’ve ever lost it at work, not even when my father was dying and I had to make a “no extraordinary measures” decision on the phone at the office. I’ve come close once or twice in some manager’s office, when I was really angry and felt I’d been treated unjustly. In that case I’d like to be treated as I just described.
My 24-year-old son did an internship working with the homeless population in San Francisco. He had to deal with a lot of much older adults who lost their composure during interviews. It was very hard for him, but I think he braved it pretty much as I described.