What material is synthesized in the rubber egg demo?
The shell dissolution reaction is the calcium carbonate in the egg being dissolved by the acetic acid, resulting in release of $\ce{CO2}$ and formation of calcium acetate.

$$\ce{CaCO3 + 2 CH3COOH -&gt; Ca(CH3COO)2 + CO2 + H2O}$$

The rubberization of the egg white is primarily a protein denaturation reaction.  Albumin in egg whites is a globular protein, meaning each molecule is a single chain that is tightly bundled into a ball-like ("globular") form.  In this form the molecules can stay nicely solvated and the egg white is liquid.  However, the low pH from the acid (and also possibly the high concentration of calcium acetate) causes each chain to unfurl and then to bind with other neighboring chains, forming a complicated cross-linked tangle of molecules.  This tangle is no longer a liquid but is a rubbery solid-like material.

So the products are primarily calcium acetate, $\ce{CO2}$, and denatured albumin.