If $X$ is the Irrational slope topology then the closures of any two non-empty open sets must intersect. It easily follows that $\Box_{n\in\omega}X$ is connected. Note that $X$ is Hausdorff but not regular. It seems (see the comments to the OP) that there are no $T_3$ examples.

If $X$ is the Irrational slope topology then the closures of any two non-empty open sets must intersect. It easily follows that $\Box_{n\in\omega}X$ is connected. Note that $X$ is Hausdorff but not regular. It seems (see the comments to the OP) that there are no $T_3$ examples.