For the 5/16 I usually run it into the refrigerator and make the splice before the water tank. For the 1/4 in I've never had one break off inside of the valve if that's what happened. If it is you're going to have to play surgeon and remove whatever is inside the valve or replace the valve. It's usually a push in. Does yours push in or does it have the plastic screw on connection? Or are you talking about the line coming from the wall? A pic would be helpful here.
You can cut the end and reconnect as long as the valve doesn't have parts of the old line stuck in it.