Meatball recipes normally include egg, because the egg acts as a binder which holds the meatball in it's shape. I have tried a few techniques for making egg-free meatballs, including using no binder at all - this is possible, but you have to be very careful when cooking them as they fall apart easily.
I have found the best alternative binder is arrowroot powder. This is often used for thickening sauces but works well as an egg replacement for meatballs and burgers. When you mix it in, you will notice that the mixture becomes quite sticky and the meatballs hold together well.
I also treated myself to a meatball making tool. It looks like a pair of scissors with cups on the end, and you use it to form the ball shape. You can of course use your hands to make the balls but this makes them all the same size.