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!
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