Why do the physical properties of an egg shell change when the egg shell is exposed to vinegar for a week?
It doesn't make calcium carbonate rubbery, it removes the calcium carbonate.  Egg shells are not purely calcium carbonate they are more like a composite with a continuous matrix of calcium carbonate and a smaller continuous matrix of protien.   When you put the egg in vinegar, you etch away this calcium carbonate matrix leaving the formerly less noticeable protein matrix as a gel filled with vinegar and creates a gel.