I got all my eggs from pink chickens, so they do count! Nothing from gnome or usual so far for me...
You don't get the eggs from completing the dead tree, I got 3 from 1 tree so there's a chance every chop! Other 2 eggs from 2 more trees, so I'm on 20% chance now...
I finished the goal getting first egg after 2 chops, so it must be getting the egg that counts for the goal!