My brother and sister-in-law have chickens. They are their "children" and they call them The Girls.

My sister-in-law wrote an answer for your question:
It takes a bit more than 24 hours for each egg to make its way down the tube, and during the journey is when the shell is formed. I also think for the most part the chickens, like birds, don't just stop and drop an egg. It's quite an ordeal I think. They definitely act broody and nasty when it's time and can barely be enticed out of a nesting box, even with food, if they are about to lay. They stand up when they finally lay. And if they are excited, they squawk and crow. If they are really broody they will sit forever on the egg. I'm sure it's all hormonal.
Best wishes.

They do. Birds, like other vertebrates (excluding mammals) have a cloacae out of which their urinal, rectal, and reproductive tracts open (mammals have just externalized the cloacae so that we have more holes). Eggs come out the reproductive tract.