You can only balance an egg on its end during the vernal equinox, which occurs near the end of March of each year.
Behind the Legend
Just to clear things up, we are talking about chicken eggs here, either raw or hard boiled (with the shell still on). No quail eggs. No ostrich eggs. Eggs that have been emptied don't count, and neither do halves of eggs or egg shells. The egg has to be stood on its fattest end on bare ground, not in the grass, not in a sandbox, not on a pile of salt. No glue, Velcro, ropes, or other foreign substances can be used. The egg must be balanced by hand, and not with a precision robotic arm or gyroscopic egg placer. The egg must remain in place for at least 10 seconds.
So, given all of these restrictions, is it true that an egg can be balance on end during the vernal equinox? The answer is yes. Can it be similarly balanced on other days of the year? Since tradition holds that you're only allowed to try it on the equinox, the answer is no.
An interesting aside, tradition also holds that if the egg is still on its end at midnight when the equinox officially ends, it is officially evil and must be excommunicated and destroyed by fire.