Unfortunately, no I don't think so. You can't even seem to store the chicken pen, either, tried that once. The good thing is that the chickens will pay you back for buying them by selling their eggs in the roadside shop (though you might want to discount them so they'll sell quickly) and using the eggs to make products (cakes, cookies are good) that you can sell for max prices.
Happy farming! Happy to be a neighbour and buy your eggs, send me a GC request.