I've got a dragon eggy :)
You need to click the eggy and load the site homepage. It's very slow though and you need to refresh it a lot at the turn of the hour in order to find any egses.
When you get enough clicks they hatch and then if you get enough clickies they grow up.
If you click in between times you can register (which can also be slow and hit and miss) then you can claim eggs on the cave screen.
I didn't get either eggy what I woz after, but just the hard rocky one. Thingy says you can collect up to 4.
Edit to add - yes, you can have 4 or you would 'overburden yourself'. Mine are not getting enough clicks, people! They'll Die!!!!!!