The story features Donald and his nephews as members of a museum sponsored expedition searching for the source of a number of square "artifacts" held in the Duckburg museum, recently revealed to be square eggs when Donald drops one and it cracks open. There is a rising interest, both scientific and financial, to find the source of these eggs and the chicken that gave birth to them. The only thing known about them though is that they came from somewhere in the Andes.
During the journey South America the nephews use the square eggs to make an omelette. This causes the members of the expedition to go down with food poisoning. By the time their boat reaches Peru the only ones that have recovered enough to continue the expedition are the youngest in the group and the lowest in hierarchy: Donald and his nephews.
Their search for the square eggs in the Andes seems hopeless as the local population sees them either as insane or as suckers to be fooled into buying artificial eggs. Finally they manage to find a very old man who can help them. He tells them the tale of an unknown man who had brought the eggs with him. The man was found by his father and himself, as a young child exhausted. He had come out of a mountain area eternally covered in mists. Despite their efforts the man did not recover and his sleep proved to be eternal. He remembers his father selling the eggs they hold in their hands.
The Ducks follow the dead man's path into the mists and after days of effort they find a populated valley in the Mountains, hidden by the mists. The bizarre-looking inhabitants of the valley speak with an old Southern American accent taught to them by their previous visitor, the dead man, the professor Rhutt Betlah (probably a play on Rhett Butler) from Birmingham who had discovered their valley during the late 19th century. During their stay in the valley, which the professor has named "Plain Awful", Huey, Dewey, and Louie produce some bubble gum balloons, which is against the law in Plain Awful. It's forbidden to produce any round objects in the valley. The only way to get out of the valley and avoid punishment, is for HD&L to produce square balloons. They manage to do that by teaching the square chickens to chew gum and blow balloons, and then hiding them under their shirts and pretending they make the balloons themselves.
Huey, Dewey, and Louie convince the very hospitable locals to let them go. The latter are sad to see them go cause they were a source of information from the outside world to their small and isolated civilization. They give the Ducks the compass that the professor had left in Plain Awful (and which was placed in a museum as a piece of art) and in turn, Huey, Dewey, and Louie teach them square dancing. When they leave the valley, in one of Barks' scarce moralizing moments, the Ducks remember that the people in Plain Awful "had so little of anything, yet they were the happiest people we've ever known."
Bringing many square chickens and eggs with them the Ducks again struggle to escape from the mists. Finally when they do manage it they are nearly exhausted and only two chickens are alive, they had to eat the rest. But it is only when they return to Duckburg that they realize the entire expedition was a failure. Both of the chickens are male and naturally can't reproduce. The story ends with Donald now giving an angry response to whoever mentions eggs and chicken to his face.
Barks had heard jokes about square eggs and chickens since his childhood and decided to use them as an idea for a story. The plot combines themes and story elements that Barks often used in his stories. A mythical creature or legendary artifact that leads to an adventure expedition, the long search for information, often seeming futile, an isolated civilization hidden from the outside world thanks to its natural environment, the Ducks bringing new ideas with them but sometimes faced as threats, and the characters ending up defeated and empty-handed are all such themes.
The story has been considered as representative of Barks' work in general and successful in its own right, and has often been declared Barks' best. References to this story are often in the works of Barks' "successors" and fans of his work in general. The author himself told in an interview in 1962 that "My best story, technically, is probably the square egg one."