MAJOR UPDATE: I just finished the whole quest(7parts) in less than 3
hours. The very end, you get the last egg(7th) along with 50 exp. and 5000gold). No more recieving eggs after that.
I found that if you give love to a burning egg on another user's farm besides Alfredo's, you will fulfill the quest and get rewarded another egg the same way you get rewarded from Alfredo's. Does it work for you, too?
So I am currently mainly giving love to users with burning eggs. I am keeping at least one egg unhatched at all times on my farm until this whole quest/event is over. Right now I have SIX Burning Eggs out on the farm for users to click on. I had SEVEN eggs so far. The first one hatched into a green hampshire. Doh! Come and visit mines to get yourself another egg. But please do the same by keeping at least one egg unhatched on your farm, too.
CHECK YOUR INVENTORY since that is where your eggs will pop up.
Let's all do the same to help each other get extra eggs/finish the quest!
Post your name/farm here and also share what you get from the burning eggs.

(If you don't confirm it working on this forum, at least post on my farm so others will know please)
For proof that it works: Check out my farm's post bulletin for users who have found it the same.
My farm: JoniVana
Eggs so far: I got all seven eggs! (Finished quest)
Hatched: 1 Green Hampshire