I'll never understand why people think eggs have so much more protein than say beans. A half cup serving of beans have about the same amount of protein as one egg (and far less fat and no cholesterol, and some beans have more calcium and iron). Tofu is a complete protein, as is quinoa. If it is the DHA that eggs offer, one egg barely offers a tiny amount of DHA, not even close to the recommended daily allowance. One would have to eat quite a few of them to meet omega 3 needs.
One point that is often ignored as far as backyard chickens is that they need constant protection and supervision, especially in urban areas. Natural and domesticated predictors abound, and chickens are nearly defenseless, especially since they have been bred for a number of years for human consumption, not for survival. Also, there have been numerous cases of hen houses burning down (with hens in them) due to house fires or overheated lamps etc. This happened in my city a few years ago. There are laws about when and where hens can be kept. They need to have the same protections as domesticated pets, but often they do not because they are considered a food commodity. And of course, as mentioned, backyard chickens had to have come from somewhere, and that probably means a farm or factory that exploits other chickens.