Isn't the egg a complex thing altogether ? We could probably find volumes of studies about it if we looked and take lifetimes to try to understand it's inner workings .
The wiki describes chicken eggshell as being .................. " 95 - 97 % Calcium Carbonate crystals , which are stabilized by a protein matrix . Without the protein the crystal structure would be too brittle to keep it's form " . They also think at the wiki that .................. " lower protein compositions are associated with lower shell strength " . Calcium Carbonate is an ingredient in some cements . So , it has a reputation for use as one ingredient in high strength materials ( relatively speaking ) .
Another source suggests that chicken eggshell is 95 % Calcium Carbonate and the remaining 5% includes Calcium Phosphate , Magnesium Carbonate and soluble and insoluble proteins . The composition of a shell is like an index of a chemistry book .
I think to bring an egg under water to apply the pressure is not what I want to do if I don't have to . I did a calculation which concludes that I must take an egg 10 metres ( give or take a few centimetres ) under seawater to add 1 atmosphere of pressure to the 1 atmosphere of pressure it would already be at , at the surface , to give me 2 atmospheres of absolute pressure at this depth . 10 metres is about 10 metres too many for me in seawater . I suppose an apparatus could be made with some hoses , a bucket with view window , some seawater and a 3 storey building to get the same absolute pressure without too much danger . Pistons , weights and seawater could be an alternative .
Again , don't injure yourself following your dreams !