Good forcings with bad squares

A self-specializing Souslin tree gives you a ccc notion of forcing whose square collapses $\omega_1$ (See, e.g., the answer to Ultrafilters preserved by $\mathbb{P}$ but not by products?). Such trees exist under $\diamondsuit$, and so are consistent with ZFC.

One can indeed prove more: starting with $V=L$, there exists a tame and cardinal preserving class forcing notion $\mathbb{P}$ such that forcing with $\mathbb{P} \times \mathbb{P}$ collapses all uncountable cardinals. See

• Adam Figura, Collapsing algebras and Suslin trees, Fundamenta Mathematicae 114 (1981) 141-147, doi:10.4064/fm-114-2-141-147.