It's very hard to tell what disease chickens may have based off of pictures, in fact there are several diseases that can only be differentiated and confirmed by lab testing. However, I would advise against eating the egg with the spot in it, as this could potentially be caused by a bacterial infection. A heavy bacterial load may also lead to sudden death as well as abnormal droppings. I'd suggest doing a thorough evaluation of each bird in the flock. Pick each bird up and feel her abdomen, is she thin, is it swollen, how does she compare to the other birds? Look at her eyes, nostrils, and inside her mouth. Is there any mucus, is she wheezing? Smell inside her mouth, is there a foul or sour smell? Look at her vent, does it appear normal, or is it inflamed and red? If possible, take her temperature. If a necropsy can be done on the dead bird, I'd suggest doing that. It could be that one bird had peritonitis or another infection in her reproductive tract and all of the other birds are completely fine. In the meantime, until you can figure out what the problem is, feed and water may include oregano essential oil and ground turmeric powder, which are natural antimicrobials. I'd suggest about 1/4 tsp ground turmeric per pound of feed and a couple drops of therapeutic grade oregano essential oil. If there is a bacterial infection, these may help lessen the load on the chickens, but even if there isn't, the turmeric and oregano will not harm them, only help them. Both oregano and turmeric can increase golden yolk color, so that's an added bonus too!
