The uniform box product is a topology, on products, between the Tychonov and the box topologies. Suppose X is the one-point compactification of an uncountable discrete space. We study the box product B and the uniform box product U of countably many copies of X. It is known that CH implies B is paracompact. Theorem1. U is normal, countably paracompact, and collectionwise Hausdorff. Theorem2. b=d implies B is paracompact.