This is a well known urban myth and it's totally false. One of the best explanations is here:
To answer your specific questions, there is no difference on the equator or anywhere else on Earth. The spin in a flushing toilet is caused by other things--not the coriolis force. If you want water to go down a drain with no vortex motion you can do it anywhere on Earth as long as you get a very big bowl, let the water settle for a couple of days so there is no motion, then remove a stopper from the bottom of the bowl without disturbing the water. It should drain straight down with no spiraling.
In future, if you hear of things that you initially do not believe, you can check at snopes, the world's best urban myth busting website I know. Here's their page on toilet flushing: http://www.snopes.com/science/coriolis.asp