  Building on work of Baldwin and Beaudoin, assuming Martin's Axiom, we
construct a zero-dimensional separable metrizable space $X$ such that $X$ is
countable dense homogeneous while $X^2$ is not. It follows from results of
Hru\v{s}\'ak and Zamora Avil\'es that such a space $X$ cannot be Borel.
Furthermore, $X$ can be made homogeneous and completely Baire as well.
