Unless you have an egg eater amongst your chooks, it does look like a crow (although possibly a rat or a lizard - If there is a rat you will find droppings) Unfortunately Crows are a highly intelligent bird and once they know where the eggs are, they are very hard to stop.
You will need to either make the run bird proof or beat the crow to the eggs. Do the hens lay in open nest boxes where the eggs can be seen? If so you could try replacing them with lawn mower catchers (openings facing away from the door) and maybe try to confuse the crow by changing the entrance
This worked for me
"Then there was a problem with the Currawongs taking the eggs so 'Chook Flaps were installed"http://gallery.backyardpoultry.com/husb ... p-cat.html