When nitric acid is added to eggshell, it dissolves eggshell. Eggshell is composed of calcium carbonate which dissolves in nitric acid. The reaction between nitric acid and calcium carbonate yields calcium nitrate and carbon dioxide gas.
$$\mathrm{CaCO}_{3}+2 \mathrm{HNO}_{3} \rightarrow \mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}+\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}$$