I don't use that type of guitar much these days, but I had the same problem on my strat. To my knowledge there is no way to avoid it happening - if the string breaks at the bridge you'll still struggle to get it out of the back of the bridge. If you wanted, I guess you could loop some cotton through the 'ball-end' on the string, and then use that to remove it when necessary.

I've always done what you did - use the end of the string, but I push it through the back and hook the string out by the ballend rather than pushing it throough from the front...