Check out the Mad Scientists Network website. It states: "Eggs do not stand on end only during an equinox. This is a popular, long-lived myth. First let's describe an Equinox: When the Earth's spin axis (an imaginary line, cutting approximately through the North and South Poles, around which the Earth rotates) points towards the Sun, it is summer for that hemisphere and winter for the other hemisphere. This is why when it is summertime in the Northern Hemisphere, it is wintertime in the Southern Hemisphere and vice versa. At two times during the year, the Earth's spin axis is 90° from the Sun. At this time the Sun is approximately on the plane of the Earth's Equator, so all over the Earth there are 12 hours of sunlight and 12 hours of daylight. These are the equinoxes - the Spring (or Vernal) Equinox occurs on March 20 or 21 (the first day of spring in the Northern Hemisphere) and the Autumnal Equinox occurs on September 21 or 22 (the first day of fall in the Northern Hemisphere).
Now back to the eggs. If an equinox caused eggs to stand on end, we would expect that there is some change in the Earth's gravity at this time. It is true that the position of the Moon and Sun influence gravity on the Earth. But there is no great change in gravity during an equinox, so there is probably no effect at all on the ability of an egg to stand on end.
This is a good science question to test for yourself with simple experimentation. Get some eggs and try to stand them on end (the larger end, that is) by first holding them up for several seconds (to allow yolk and whites to settle at the bottom), then letting go. Do this with several eggs and record how many eggs stood up and how many fell over. Also record the date. Do this experiment at different times of the year (including equinox and non-equinox days). Then compare your results from different times of the year. I'll bet that you'll always find that some eggs will stand and some will fall, no matter what time of the year it is."