Hi!
The cell structure that most resembles the function of an egg shell would be the cell membrane.The cell membrane is to a cell what an egg shell is the the egg.
The egg shell allows particular substances such as air and moisture to pass through which are required by the egg for proper development, similarly the membrane allows selective substances to pass through it which are required by the shell, and does not allow other substances that may be harmful or unnecessary for the cell to pass through.
The egg shell is a semi-permeable structure, just like the cell membrane.
Hope this helps!