I think you might be right about the eggs giving Hayners. The bonus time is, once again, such that it's 30 minutes super hard bosses and then 30 minutes of "rare enemies", which, from what I experienced, seemed to only increase the number of egg enemies in any given level. There weren't any adventurers that would automatically drop Hayners or anything. It's the same missions as any other time (outside the lvl 500 boss times), except there were quite a lot more eggs around each time, at least in some rooms.
And yeah, it DOESN'T work all that well, since you don't get the treasure unless you're participating in the battle against one. Reeeal shitty way of giving these things out, since it's so hard to coordinate it so that every rando would participate in an egg battle before it's killed.
