Assuming C/C++, dare we even ask what this teacher uses instead of switch statements? Or are her switch statements unreadable rat's nests of extra conditions?

This is a good life lesson. We're all idiots about certain things. Your teacher, me, and even you. It's even possible to be a recognized expert in a field yet still be an idiot about some particular thing in that field.

Just because some people use a screwdriver as a hammer and risk injuring themselves and damaging their work, that's not a good reason to insist that no-one should ever use a screwdriver under any circumstances, is it?

Use break statements when they're appropriate. Don't use them when they're not. Learn the difference from code that many other people recommend, like popular open-source libraries and tutorials. If there's a preponderance of break statements in your code, you may be using a suboptimal approach.

But unfortunately, for this course, your best bet is to nod, smile, and not use any break statements. Look at it as a personal learning experience; by forcing yourself sit down and reason out how you can do something without using break statements, you might find some situations where they weren't actually the best solution. And when you can honestly look back and say that the solution with break statements is objectively better, you'll be able to use that approach with greater confidence in the future.