I've managed to resolve the question.

Regarding the Box Topology the claim is correct directly from the definition of the open sets in that topology. Since every interior is an open set in the according topological space the product of the interiors is a basis element of the box topology and thus open itself, meaning it is the same as its interior.

Regarding the product topology the claim is incorrect. I proved this using a small llema stating that the interior of the product is an empty set unless apart for a finite number of indexes it holds that .

Thus is suffices to take as a counter example since

is clearly not empty.