Your professor was saying "breaks should be avoided" because they are similar to the now-taboo goto statement. However to say "breaks should be avoided whenever possible" is just wrong.
Break exists for readability. You can do anything without break that you can with it, but that doesn't mean you should. Sometimes it just makes sense to make a statement like "In this condition, break out of this loop." or "In this condition, return from this function."
It's a bad idea to use break for program flow in a nested for loops in a way that might give a casual reader some confusion.
Simple rule of thumb: Does using break here (instead of additional if/thens, another boolean, etc.) make this easier to understand or harder to understand? It's that simple.