The uniform box product problem is a weakening of the well-known box product problem, which asks whether box products of certain compact spaces are normal or even paracompact. Using uniformities, a new topology on products is defined between the box and Tychonov topologies. This new product, called the uniform box product, is an extension of the sup metric to powers of compact spaces. We investigate a certain non-metrizable compact space whose uniform box product, in ZFC, is normal, countably paracompact, and collectionwise Hausdorff.