There is a widely popular myth that the vernal and autumnal equinoxes are the only times of the year at which you can stand an egg on its end. The basis appears to be that during this time of year, all the gravitational forces from the Sun and Moon balance out perfectly as the Earth stands with its axial inclination at 90 degrees from the Sun, giving the Earth equal night and day lengths.
The truth of the matter is that the gravitational pull of the Sun on the egg is thousands of times smaller than the gravitational force of your own body on the egg, due to being a few million miles farther from it (gravitational force decreases with the square of distance). Furthermore, the Moon's pull is a random wildcard, since the relative position of the moon has nothing to do with the equinox and it could be any any side of the Earth-Sun system. Fortunately, its gravitational pull on the egg is also so small as to be insignificant as well.
It is, however, true that you can balance an egg on its end at the equinoxes. It is also true that you can balance an egg on its end at any arbitrary time of year, but most people don't bother actually trying it outside of the equinox so they don't know that. Raw or hard boiled, it doesn't matter. All you need is a steady hand and a little practice. I did a demonstration of this once at the summer solstice, just to prove the point.
This is a classic demonstration of the power of the scientific method in action. The scientific method demands that a hypothesis be tested, multiple times, and under different conditions, in order to eliminate random environmental variables and confirmation bias from skewing your data. Most people, as I said, only test the egg balancing hypothesis on the equinox. Upon success, they declare the hypothesis proven, but it isn't! Without attempting to balance the egg on a day other than an equinox, nothing has been shown about conditions under which the feat is impossible. All we have is a single data point.
There are ways to cheat, of course, if you are so inclined. A borderline case is to shake the egg first. This breaks the yolk loose from the bands that keep it suspended in the albumen, letting it settle to the bottom, giving it a lower center of gravity. Another method is to set the egg on some salt and then gently blow away the grains from around the egg. This leaves a few grains of salt around the base, almost invisibly helping to hold the egg upright. And then there's the apocryphal story of Christopher Columbus cracking the end of a hard-boiled egg to flatten it, but that's more of a lateral thinking exercise.