No. A subforcing of a c.c.c. forcing is c.c.c. A subforcing of a countably closed forcing is countably-strategically-closed, which implies proper. (This is easy to see via countable elementary submodels. Use the strategy to construct a generic condition.)
Furthermore, every subforcing of a proper forcing is proper. Properness is equivalent to preserving stationary subsets of $[\kappa]^\omega$ for all $\kappa$. The stationarity of $X \subseteq [ \kappa ]^\omega$ cannot be restored once killed, since killing it is just adding some $f : [\kappa]^{<\omega} \to \kappa$ such that no $x \in X$ is closed under $f$. (See Abraham's chapter of the Handbook.)