To keep your chain from breaking, keep the derailers properly aligned, ease off of pedaling when you shift, and don't "stomp" on the chain when it's not fully in gear. (Folks who don't go back to the non-indexed days probably never got into the habit of coasting while shifting, but it wouldn't be a bad thing to practice a bit.)
Broken chains are exceedingly rare, and are almost always the result of putting load on the chain when it's not fully in gear. The other likely source of damage is having sprockets that are so badly hooked that the chain gets sucked into a knot.
Update -- 6/14/18
An observation I made this week, working on a bike for Christmas Anonymous:
The bike (a 26" full susp BSO) came in with a mangled chain, a mangled rear wheel (half the spokes were broken), and a broken rear derailer. The chain had actually been twisted 90 degrees at one point (amazing that it didn't break), something I'd never seen before. It seemed like the chain had gotten twisted up in the rear cluster perhaps, after the rear wheel had gotten mucked up. (Perhaps a garage accident -- hard to tell with such things.)
So I found a new rear wheel, a new rear derailer, and a new chain, and put it all together. The rear derailer worked fine, once I got it adjusted (involved some cussing), but then as I tried to get the front derailer to work (blasted twist shifters, plus a cable that was routed so that it collected rainwater) I started having trouble with chain suck on the small front ring. I could see that this chain suck likely caused the chain to knot up and twist, and it was only after that that the garage accident occurred, taking out both wheel and derailer.
Since the chain suck was consistently on two teeth a one point on the small ring, I was able to use a round file to smooth out the teeth and eliminate the problem. I'm guessing the teeth had burrs from when the ring was manufactured, and the bike never got used enough to wear them off.