Not a disaster, as Simon says, the good news is that the bridge hides everything, so it really doesn't matter what you have to do to get them out. Within reason you can butcher it until you are ready to cry at the mess and no-one (except us) will ever know.
Easy outs/broken screw extractors can work, but there's a certain amount of voodoo about them. I think they are one of those tools where practice helps, and the first time you use them it goes wrong. If you have a mate who has used them successfully rope him in, but I'm not sure I'd bother buying a set yourself. They work much better if the screw head is just rounded off rather than broken.
To be honest I'd probably get something like a 2mm drill bit and drill holes all round the broken screws as close as I possibly could so there's nothing left for the screw to grip on, and then chop away enough wood near the surface to get a grip on the screw end with pliers. If you can drill down the centre as King Casey says that can work too, but it can be very hard to get the drill to stay in the centre of the screw without a pillar drill. Still, if it slips off to one side that's only one of the surrounding holes I suggested. Then when whatever ghastly mess has ensued, just drill the holes out bigger to whatever size of dowel will fill them, glue in the dowel, probably titebond or similar, then you can start again with good quality screws (i tend to use boat shop stainless steel) and a size larger guide hole... I had to do this recently with my first bass, which had one broken screw and others pulling out, and you'd never know to look at it.