Hmm. I wonder if they would be willing to do without their electronic toys until they "bit the bullet" and figured out how to accomplish the task on their own? What I meean is, you say you've done it with them, you've "held their hands," so to speak, you've encouraged them, provided protective gear (gloves), etc., now you think they should be able to do this on their own. So ... what if you tell them you're confident they're able to solve the problem on their own (without harming Ms Grumpy Beak, of course) and that when they do, they can come in and play their video games. And of course you have a pretty good idea how many eggs to expect and can certainly go out and check to make sure they did indeed gather eggs from under that chicken, since they did lie to you before. If they do lie again, simple.. they can spend the evening reading (we call these things books, children) and not get their devices. They can try again tomorrow. You can wear earplugs to protect yourself from the whining. 
