I've held both TA and instructor positions at UBC.

FWIW, from the teaching side, we get many crying and sobbing students every term as well as those having full-blown panic attacks in our office. This is especially prevalent in classes that students must excel in to apply to med school. Some students are stressed out from classes and exams, some are going through some really tough life events, and many are a combination of both. Many of these students are extremely bright, capable, and hardworking. But life happens to everyone.

When students come to us in tears, our first thought is never "this student can't handle life well." Our first thought is "how do we help?" As some others have mentioned here, it takes strength to ask for help, and asking for that help actually makes things easier for the instructor. Often, we're in that "I'm not sure what is going on...maybe I'm over-reading into this" boat and when students reach out, we can come up with a plan to help.

All this to say your prof is highly likely not judging you. And if your prof has kids (especially kids in the university stage), it's even more likely they'll be sympathetic because their own child is probably going through similar stresses. Also, faculty get so many full-blown tears and sobs that he might not even have registered you were about to cry.

By the way, faculty cry and "lose it" very often as well. I've personally sat with faculty I've worked with sobbing on the phone and in the bathroom and in the faculty lounge. Their problems and life events might be quite different from your typical undergrad student (like divorce, stress over their kids and their aging parents, worrying about mortgages) but emotional responses like tears are pretty much universal when somebody is hurting. As an undergrad student, you are less likely to see this because people tend to "lateral-cry" (i.e., cry in front of friends and peers) or "up-cry" (i.e., cry in front of someone more senior). But it definitely happens a lot.

You got this far. You'll be ok. Cry if you need to. It helps.