The eggs are part of the Chicken Coop collection. These randomly drop when you build a chicken coop and when you collect the daily bonus from a finished one. The goal hints that you probably need to build a second one to find all the eggs you need to complete the goal.
Building a new coop means you need to collect 15 pieces of each building material type. This is not an easy task. What you can do is to purchase a new Chicken Coop from the market but don’t complete it. Just whack the frame until it’s time to complete it with the materials.
Whacking the Chicken Coop can drop eggs needed for the Collect a Dozen Eggs goal. Once you’re done with the whacking part, delete the incomplete Chicken Coop and buy a new one. Repeat doing this until you complete the goal.