Does there exist a context-free language $L$ such that $L\cap L^2$ is not context-free?
You can take
$$
L = \{ a^n b^n : n \geq 1 \} \cup \{ a^k b^n a^n b^\ell : n,k,\ell \geq 1 \}.
$$
You can check that
$$
L \cap L^2 = \{ a^n b^n a^n b^n : n \geq 1 \}.
$$