Rub a little graphite (from a pencil lead, say) into the notches in the nut and anywhere else where strings may be catching. You might also use the little tubes or pads that you'll often find attached to strings; they are intended to ensure the string's easy passage over the bridge.

A string breaking between peg and nut is likely catching at the nut; the slot may be too tight or not deep enough, but I'd try graphite for lubrication before anything else.