I can confirm that the quest is bugged, I had to abandon to get the quest. Here's how it worked for me.
Got an egg at around 10pm last night. Abandoned it. Couldn't initiate it again until 4am. Abandoned it, then I just initiated it again at 12pm today. Eastern time zone.
With another person giving me some eggs (and me wasting gil in the wards) I got the Pristine Egg Cap.
I assume that in the future when you complete The Dreamer's Gospel in one of the states, it unlocks the next tier quest for the next tier Egg Cap (they go r10, r15, r20, r30, then r40 if I remember right). Maybe?
