  We present a new application of the squeezing function $s_D$, using which one
may detect when a given bounded pseudoconvex domain $D\varsubsetneq
\mathbb{C}^n$, $n\geq 2$, is not biholomorphic to any product domain. One of
the ingredients used in establishing this result is also used to give an exact
computation of the squeezing function (which is a constant) of any bounded
symmetric domain. This extends a computation by Kubota to any Cartesian product
of Cartan domains at least one of which is an exceptional domain. Our method
circumvents any case-by-case analysis by rank and also provides optimal
estimates for the squeezing functions of certain domains. Lastly, we identify a
family of bounded domains that are holomorphic homogeneous regular.
