How and where to find Ant Eggs

Ant eggs can be found in Ant hill north of the initial spawn point. 
Inside you will find a big cavern filled with red grass, that is the place where ant eggs will spawn.

If eggs are not spawning for you you will need to kill more ants so they start laying eggs.

Direct quote from one of the developers: 

"The ant eggs spawn based on how many ants are alive. So if you aren't seeing any ant eggs in game it might be time to take out a few ants."

If left alone Eggs will hatch after a while.