If $\mathcal{C}$ has finite products, then the class of projective cofibrations is also closed under finite products. Indeed, since the cartesian product in the category of simplicial presheaves preserves colimits in each variable, it suffices to check the claim on the generating projective cofibrations, i.e. that for every pair $(A, B)$ of objects in $\mathcal{C}$ and every pair $(n, m)$ of natural numbers, the inclusion
$$(\partial \Delta^n \odot h_A) \times (\partial \Delta^m \odot h_B) \hookrightarrow (\Delta^n \odot h_A) \times (\Delta^m \odot h_B)$$
is a projective cofibration. But this is isomorphic to the inclusion
$$(\partial \Delta^n \times \partial \Delta^m) \odot h_{A \times B} \hookrightarrow (\Delta^n \times \Delta^m) \odot h_{A \times B}$$
and $\partial \Delta^n \times \partial \Delta^m \to \Delta^n \times \Delta^m$ is a cofibration of simplicial sets, so the claim follows from the fact that $(-) \odot h_{A \times B}$ sends cofibrations of simplicial sets to projective cofibrations of simplicial presheaves.

A similar argument works for projective trivial cofibrations.