Theoretically a lot of stuff can happen. As a practical matter, this has to be regulatory. Eggs and chicken meat is a very well understood product and by nature a completely normal market. I've literally been offered chicken meat by roadside sellers who butcher them on the spot and cook up a meal; it is a bit gruesome but not outside the complexity of operation that one family can manage.

The part of the supply chain acting as a "monopoly" is too simple to replace. Someone can literally set up a new local butchery, buy directly from the farmers and the exclusive dealership network is broken. Probably need some know-how and a refrigerated truck. It is barely possible that the sort of monopoly structure being described could have been built over the entirety of the US without regulatory support. It'd require something approaching a mind control ray to keep the competition under control.

I believe the other comment saying that chickens are a bad business, competition would tear through the margins like a rooster eating a mouse. But a country wide monopsony on chicken purchases is absurd (unless the regulators are pushing it). If nothing else there are a bunch of low-income people who'd be happy to buy and butcher their own chickens if it cut out a middle man. It isn't feasible to build a natural monopoly.