My kettle has a boil screen on the dip tube. I know it's not the same thing as a SS braid, but the only time the boil screen has clogged was when I did a pumpkin ale and had a ton of pumpkin in the kettle. I haven't tried it w/ whole hops or plugs; I use pellets in a mesh bag and have never had a problem. But if you just throw your hops in the BK & combined w/ the break, I agree w/ BargainFittings you're likely to clog the braid.
As for the break, I don't worry about getting some in my fermentor, because there are nutrients in the break that are good for your yeast, especially zinc. Heck, lately I don't even bother w/ a secondary. I'd rather not waste the beer by leaving a quart here, a quart or 2 there. I've never noticed any problems w/ my beer. I guess the only time I'd worry about getting a lot of break and hop material in my fermentor was if I was doing a really pale and/or low gravity beer.