I am familiar with the result that $\Diamond$ implies that there exists a Suslin tree, but does it also imply that, if S, T are Suslin trees, then S x T is a Suslin tree? If not, perhaps there is a special class of Suslin trees where this nice result holds. Any input is welcomed.
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This is an interesting topic. The result is that you cannot really say much about $S\times T$ in general.
To be more precise, $\diamondsuit$ allows you to construct trees $S,T$ that are Suslin, but their product is special, or $S,T,U$ that are Suslin, $S\times T,S\times U,T\times U$ are Suslin, and $S\times T\times U$ is special. In fact, just about any pattern is possible.
An excellent reference for this and for techniques to use this freedom to "code information" is the paper
The result is false in general since $S\times S$ is never Souslin. However, you can use $\diamondsuit$ to construct Souslin trees $T$ and $S$ such that its product is Souslin. You can find the construction in Todorcevic's paper in the Handbook of Set-Theoretic Topology.