Are eggs the menstruation of chickens?

Absolutely not. First of all, menstruation only exists in primate mammals. They are haematic and tissue losses due to the superficial flaking of the endometrium, the tissue that covers the inner cavity of the uterus and that changes during the female menstrual cycle whose tissue breakdown results from the failure to fertilise the egg. The hen’s egg is produced in the ovary from which through the infundibulum it passes into the magnum, where a large part of the albumen is formed; then it stops in the isthmus where the testaceous membranes are formed; then it passes into the uterus where it is covered by the shell, then through the sphincter to the vagina and then into the oviduct and finally laid regardless of whether fertilisation has occurred or not.

So the egg is not the hen’s menstruation, because menstruation is the breakdown of uterine endometrial tissue in primate mammals in response to the failure of an egg to fertilise, whereas the hen’s egg is the end product of a series of steps unrelated to the fertilising act that, if it occurs, allows the embryo to develop outside the mother’s body.