BYC, this idea isn't unique to us. I just did a quick Google search of "chickens in the White House" and found many links to articles about others with our same idea. It may only be a matter of time before there's Presidential peeps scratching up the Rose Garden.
I also did a quick search on Presidential pets, didn't find any chickens but did see that President Wilson had a flock of sheep grazing on the White House lawn, and his predecessor Taft had the last White House cow, named Pauline Wayne.
I imagine there have been chickens kept by the White House kitchen back in the day when yardbirds were the custom for fresh eggs & meat. But perhaps not kept as pure pets.
They do roll eggs on the White House lawn every Easter, it should be an easy progression to go from eggs on the grass to keeping chickens there.