There is a basket of eggs upon the table. The performer takes an egg from the basket, places it in an egg-cup, and covers it with a hat. He takes a handkerchief and scrunches it up between his hands. When it's no longer visible, he opens his hands to reveal the egg, then removes the hat to show the handkerchief. The egg and the handkerchief have changed places!
At the back of the basket is a wooden half-egg, painted egg-colour on the outside and the same colour as the inside of the egg-cup on its inside. It is exactly the right size to sit on the egg-cup or fit inside it. It contains a handkerchief exactly like the one that's scrunched up. It's this that the performer actually uses rather than one of the real eggs. When the performer covers the egg-cup with the hat, he turns over the half-egg so that the handkerchief can be pulled out at the end of the trick. When he seems to be scrunching the handkerchief up, he's actually squeezing it into a metal or plastic egg, of exactly the right colour, which has a small hole in one side. This he palms and presents as the original egg, and then lifts the hat and pulls out the handkerchief.
Nothing difficult about this trick except for the manual dexterity and misdirection required. The metal egg can be concealed anywhere, even within the handkerchief when it's first picked up.