I had to look up the bouncy egg experiment, but it seems like you submerge an egg in vinegar and after a while you can peel off the shell and the resulting egg is bouncy (if you are referring to a different experiment let me know!).

The reason the egg dissolves is also the reason it bubbles while dissolving - the acetic acid (CH₃COOH) in the vinegar reacts with the calcium carbonate (CaCO₃) in the egg and produces Ca(CH₃COO)₂, water (H₂O), carbon dioxide (CO₂). The bubbles you see on the egg shell as it dissolves are bubbles of CO₂ formed by the chemical reaction!