All animals~and humans, for that matter~deserve a little rest in the winter months. The trees are dormant, plants are dormant, bugs dormant.....why is it that folks expect the chickens to go on shooting out those eggs like gumball machines when the rest of nature gets a few months off? Tell him you need those mature hens on hand when you save one of those roosters to mate with them, then they can replace themselves~cheaply~so you won't be in this fix next winter. From now on let the old hens raise up some new ones each year and you need never be in a pinch for eggs again if you time it right.
Tell him that he is, quite literally, throwing the baby out with the bath water if he kills your hens right when they are on the verge of making replacements for you guys. Then start fermenting the feed so you can cut that feed bill by about half...that should make him happier.