What is the chemical composition of an egg shell?
It's the same stuff that makes up most sea shells, calcium carbonate, $CaCO_3$.

Incidentally, this explains why egg shells dissolve in vinegar (acetic acid which, since it is an acid, provides the Hydrogen ions in the reaction below): 

$CaCO_3+ 2H^+ -&gt; Ca^{2+} + H_2O +CO_2$ 

This simple reaction (which is what would happen in your stomach as well) produces water, calcium ions and carbon dioxide, none of which are particularly harmful.

The specific reaction with acetic acid is

$CaCO_3 + 2CH_3COOH → Ca(CH_3COO)_2 + H_2O + CO_2$

which also produces calcium acetate.

Anyway, there is nothing particularly toxic about $CaCO_3$, it's basically chalk. I wouldn't recommend eating a lot of it but you shouldn't freak out if some falls into your omelet either. 

However, don't forget that an egg has exited a chicken's anus (well, cloaca) so their shells are likely to have come into contact with feces and they are porous so they are not necessarily the cleanest thing around. This is just not because of any chemical toxicity.