Re: 1958 Johnson Seahorse FD-12 18 hp
Let's just assume the eccentric screw didn't break because it is corroded in the hole and wouldn't turn at all.
As you know, the screw is threaded. As installed, it is turned all the way in, then backed off far enough to allow back and forth rotation to move the breaker base. What happened is that yours was run in all the way and not backed off. So when you attempted to turn it a bit more to adjust the breaker, you snapped it off.
It should be easy to remove the broken piece if you can just get ahold of it somehow. I suggest removing the armature plate and attack the screw from the bottom end with a small drill bit. The bit probably will zip it right out. However, forget it if it is broken because it is sized in the hole from corrosion.
If all else fails, you don't really need that screw. It doesn't do a darned thing except make it easy to set the points. Once they are set, the eccentric does nothing at all. What I'm saying is move the breaker base manually to get the gap setting and let it go at that.