The Broken Egg Shells is a piece of evidence that you will need to retrieve to prove the existence of Cryptids which is your main mission in the island. These broken egg shells are from the nest of Jersey Devil.
How to get Broken Egg Shells
Once you got the Grappling Hook in the house, exit to the house. You will see the Jersey Devil flying and you will notice the nest on the tree. Use your grappling hook and it will hook to the nest. Climb up and get the Broken Egg Shells on the nest.