Love my dogs too. I can call my dogs off ( obedience trained) but how they behave when I am not there is another thing. I don't have this problem but dogs are forages and I'm sure coyotes make a meal of tasty turkey eggs ; but I did think up a possible solution.
Try setting the dog up for a bad experience. Set up several specially prepared eggs where the dog can get them. Inject a good tablespoon hotsauce inside the eggs. See how many the dog eats. If the dog quits and this seems to work, sprinkle a few drops in the nest around the eggs. This will remind the dog as he is sniffing around, of the aweful experience he had eating eggs.
Learning can be thru one very bad horrible experience, or a series of repetative training efforts.
I don't eat raspberries with cream anymore! I ate two bowls and spent the next 24 hrs throwing up. THat was 25 years ago! One bad experience . . . .