How to stop thrown eggs from spawning chickens?
Solution 1:
Just random ideas:
Ender pearls. You can empty the thrower tag to not teleport players around. But I don't know if ender pearls fly farther than snow balls. In that case one team would have an advantage.
Custom ground/hit detection: Depending on what your game needs are, you could detect if there's a block closely under the egg or a player nearby or whatever. But that could be hard to do without strange side effects (like hitting when in reality missing barely).
You could combine the custom hit detection with teleporting a pearl or arrow to it (I think that copies the motion, I'm not sure, it's a long time ago that I checked) and kill it after a few ticks if it didn't hit. That way it would hit the ground/player and hopefully avoid the disadvantages of 1 and 2.
Make chickens invisible with a texture pack. If that works in your map. If you don't need chickens anywhere else, this is the easiest and best solution. Only F3+B would still show them for a frame.