So, after winning a clan battle, you have chance of getting an egg. This screenshot is taken from discord. For that, you need to be in a clan. Simply head over to your clan page (you can do that by clicking your clan name at top left) go to clan battles and battle your opponent. There are some clans recruiting currently. You can join the official pokemon vortex discord for more info.
Here is the invite link: https://discord.gg/pokemonvortex
Another way to get eggs is to purchase them from auction.