Since egg shell is made of calcium carbonate. And we know that when calcium carbonate (CaCO₃) reacts with nitric acid (HNO₃) then carbon dioxide, calcium nitrate and water are formed. So same reaction happens when nitric acid is added to an eggshell.

CaCO₃ + 2HNO₃→ Ca(NO₃)₂ + CO₂ + H₂O