I have a Hobie 16 and the gudgeon pulled off in the last race. Three screw heads broke off leaving the screws still mounted into the transom. I managed to get 2 of them out, but one still remains. I was able to drill into it, but the drill bit broke off inside the screw. I ruined 3 more drill bits just trying to drill through the broken one. The bolt is broken off at the aluminum plate about 5/8 deep inside the transom. It's not coming out. Based on the other 2 that I extracted, there is probably very little of it protuding (on the inside) past the aluminum plate, so I think that porting the hulls and extracting the screw from the inside would be in vane. It would be MUCH easier to just move the grudeons over a 1/2 inch and redrill and tap the holes.
Main Question: Would I still be race legal?
Second Question: Should I use the one piece gudgeons instead? (I would still need that same hole, but the new gudgeon has eight screws (in my case seven) instead of the standard 6.)
Whooo, boy! You're screwed! 
/thanks, I'll be here all week. Try the veal!
High road - go with the one piece gudgeons. They work really well.
Low road - forget about the one screw. Drill and tap the other holes to accept 1/4" machine screws and use only three in the lower gudgeon. Should hold up fairly well.
Is there some kind of backing plate in there, or just thick fiberglass the scews go into? You may be able to put a small (read: very small) inspection port behind the back beam...Then take a hammer and a punch, and just knock the screw into the hull. Be careful not to damage the transom any more, and then glass in the inside of the transom, retap larger screws, and keep on truckin....
Trey
N20 314
Layline Rigging
There is an aluminum backing plate in the transom. For one bolt on the 4 bolt gudgeon, I wouldn't cut in a port.
Cool Picture.
I put the "Euro" one piece casting on. The screw holes are slightly different (good)but the holes still picked up on the aluminum plate when I drilled and tapped them, and I bedded it in some thickened epoxy, so then, really, there is not much load on the bolts. Worked for ever, no problems.
Had also broken the stainless gudgeons (one broke, one cracked)
How did you not trash the fiberglass lip at the through hole when it ripped off?
I'm going to use the newer one piece gudgeons and follow the advice about using the larger screws. I'm alittle concerned about re-drilling the holes. The holes for the bracket and the existing holes seem to be misaligned (bracket is metric spacing and boat has non-metric spacing) both vertcally and horizontaly. Everything is off by about 1/16th. I'm afraid that the drill may "walk" toward the existing hole instead of going straight and enlarging the hole in the proper area. I need to put my boat in a drill press! Too bad Hobie didn't take the time to create a US version. >:-| I'm also going to inject some epoxy resin to fill any voids. Thanks for all the advice. I'll repost when I finished (may take a while).
The fiberglass lip wasn't trashed because the top gudgeon still held (surprisingly!) I was trying to figure out why things were so sloppy. I looked down and the rudder was slanted pretty badly. That's when I knew that I was done for the weekend. Hey, at least I got to sail a little bit.
Here's a bit of a trick that will keep you from having to put a port in:
After you mark the location of the holes, put one of the old bolts back in to hold the backing plate securely while you drill and tap the new holes. Otherwise, you could end up pushing the backing plate into the boat. 
I learned this the hard way on my 17 - which now has another port in the deck ahead of the transom. 
On the other hand, putting in ports is a good thing. When I put two ports (the maximum allowed by class rules, at least at that time) in each hull, and then left them open full time (covered from rain but letting air flow through) each hull lost some weight (7-9 lbs per hull) and ultimately it was better for the boat.
I am now a firm believer that any boat that doesn't have portals should and I have added them if the boat doesn't come with them.