There are rumors that people can balance an egg on its sharper curves during an equinox (something about the spin of the earth and the position of the sun will let it stay up). That's crap. In order to stand an egg upright you have to control the spin of the egg. Which is good, because that's a lot easier than controlling the spin of the earth.
To start with, the egg has to be boiled. Raw eggs, when spun, will wobble back and forth on the table as their yolks move inside them. Only a boiled egg will have a regular spin. The ideal spin is anything over ten revolutions per second. Most people are able to pull this off with a sharp flick of the wrist. The egg will wobble, then push up and stand on one of its sharper ends.
It took mathematicians six months to work out equations for how this flip worked. It starts with the symmetries and asymmetries of the egg. When the egg just sits on a surface, the center of mass is balanced over the single point that touches the ground. When it begins spinning, it is spinning with its shorter, blunter end on one side and its longer, sharper end on the other. Instead of spinning on one contact point, it lurches to the side and drags that single point of contact around in a circle as it spins. That circular motion creates a large frictional force; not enough to stop the egg, but enough to push it around and up.
As the egg gets pushed up, it gradually switches from an axis of spin around which it is asymmetrical to one in which it is symmetrical. This stops the point of contact from being dragged around in a circles, which stops the frictional force. The egg doesn't "defy" gravity. The force of gravity is just less than the force of friction pushing the egg into a different rotation. As the above video demonstrates, it even works with Creme Eggs.
And yes, you can balance eggs on their points without spinning them on the equinox. It's just that you can do it on any other day, too.