How do I quiet down the egg song? Is there a way to sound proof the coop while still having ventilation? Any way to move it later in the morning? Right now it's happening every morning at 6:30AM, and I suspect my neighbors will soon hate me.
6:30 am! Are they laying at that time? Or just "going off" like alarm clocks?
The 'egg song' can indeed be stressful.
I personally believe it's the result of thousands of generations of people popping up right while a hen's on the nest and violating the secrecy, security and sanctity of the nest to snatch eggs before egg eaters or whatever takes the eggs... Because you may notice, the so-called 'egg song' is actually identical to the alarm/panic call, the one they use when a predator is spotted. Not the 'potential threat' vocalization, either, it's the full blown 'predator attacked' noise they make which we call the 'egg song'.
What wild species naturally spends time alerting all enemies and predators to the location of their nest or offspring? Some wild birds sing within their territories, but not ground-nesting ones, and they don't make a ruckus at the site of their nests. They move away.
At any given time I would have a minimum of 50 hens and an approximate amount of roosters, of all manner of breeds all mongrelized together, and the 'egg song' was never something I heard.... Until I got in a hysterical hen who loved the sound of her own voice. So she would start, and with that edge of panic in her voice, the roosters paid attention and tried to spot the source of distress. Failing that, they assumed she wasn't crying wolf, and joined in. So did all the other hens. Within a week or so, I had a whole yard and paddock full of over 100 chickens singing the Egg Song all day, every day. NON STOP. It was maddening.
The trick was to attack the ringleader/s. When those most keen to begin it started, I singled them out, and made as if I was going to catch them, but I gave them vocal warning to shush. After a few repeats of this they understood, and soon, if they began to start the ruckus, I would fix them with a beady eye and give them the order to shush. Sometimes I'd clap my hands to reinforce it. Body language counts for a lot too. Basically, within a day of being consistent I had about 99% of the flock calm and silent again, but one of the hens didn't want to stop, and I removed her. As in, ate her, but really if she was a beloved pet I would have tried other means first. She wasn't, her loss. LOL that may sound harsh. But I don't look favorably on hysterical or stupid animals. It breeds on in most cases. I could have tried using a water gun on her... But I had a big yard, big paddock, and didn't want them to learn to run from me... But a method like that may work better for you.
Chickens are trainable, and they do learn from watching (new scientific studies 'reveal' the ancient truth, that animals do learn from watching one another). So tackle the one that starts it. If push comes to shove, remove her. Do not be fooled into the trap of giving them treats to distract them. Just like any other animal do not reward negative behavior or you reinforce it. If they find treats rain from the sky whenever they start yelling, they will only do it more often.
There's also a chance something they're spotting in the early morning is frightening them. A neighbor's cat perhaps goes through your yard, maybe someone walks their dog past, maybe you have a resident lizard or snake that comes out to sun itself. Could be anything.
Best wishes.