Ant Eggs are used in a couple of different recipes, so if you want to blow up a bunch of bugs, then you're going to need to have some at your disposal. You can make two different explosives with it, one of which requires Ant Pheromone, which you can craft with the eggs. So, these are pretty useful to have if you really want to do some large scale destruction or knockout some ants.
You can get Ant Eggs by taking a trip down into the anthill that is located to the northwest of the first Field Station. If you can fend off the Ant Soldiers, you will find the eggs in a couple of different small areas.
As you head down the anthill, you should hug the wall to the right. You will eventually end up at a small point where eggs can spawn. It looks like this:
If you keep going further down that path you will run into the second room where eggs will spawn. This room has a hole in it that you can go down to reach a lower level of the nest.
While you're in this area, keep an eye out for skeletons. You will find old bee armor near them that you can bring back to your base and analyze for some science!
If your Ant Eggs are not spawning, according to the developers themselves, all you need to do is slay out a bunch of ants to get them to spawn. If you reduce the population of the ants, then they will be forced to make more eggs to replenish it. You can then go to the nest and grab some eggs! This can take a couple of in game days to occur, so you need to be patient. This can be hit and miss as I've seen different reports from people having luck with it and others not finding additional spawns. If it is a bug, hopefully they get it fixed in the next update.