When you see the red “X” icon over the eggs in a Farm Kitchen recipe menu, there should also be a blue “Select” button underneath.
Clicking the “Select” button will take you to the Settings > Eggs Priority menu.
Although you may have eggs in your Barn, they’re most probably the egg types in slots 4 & 5 in your “Eggs Priority” menu, which are the egg types that are saved.
Since only the first 3 slots of egg types are used by your Gear and Kitchen, click, hold and drag the egg types to reorder them and set which ones to use and which ones to keep.
You can also hover over an egg type to see how much you have in your Barn.