How to keep a screw from slipping off a screw bit

The fact it broke off means the screw is probably very stuck. I would first use a penetrating oil (though WD-40 might also work), and let it sit for awhile, before trying anything further.

Hindsight is 20/20 of course, but it would have been good to do this first. If you see any corrosion or rust, or suspect it's needed and have time, this never hurts to do.

You may be able to use a small drill bit to extract the broken extraction bit, and then try again -- but be careful as you may damage the screw beyond my next tip..

If you can't get the tip out (or don't want to risk trying), I would probably use a dremel cutting wheel, and basically turn the screw into a slotted screw. This can be difficult -- not deep enough, and you'll just strip the screw worse trying to get it out, and too deep and you'll weaken the head so much it'll probably snap off when you try to turn it.

The next thing beyond this is to use a "reverse drill bit" or "left-handed bit". You can drill right into the middle of the screw itself (with your drill in reverse), and the bit should actually come out before you've gotten all the way into it. Use a bit that is smaller than the thread size, otherwise you'll strip the threads so you won't be able to put another screw back in. Again, judgment is necessary here, too small and it won't get enough grip, too big and you'll strip the threads -- though maybe by this point you don't care that much :)

The best way to get it out is muscle. Not much else. Use a vice grip type plier to get it out. If it breaks off go to the other side. Some people say tighten it up into a drill and use the drill to get it out. When you get it out then re-tap the screw hole. In the U.S. electrical boxes are 6/32 and most DIY's and electrical wholesalers sell taps for this reason. Good Luck.