Suppose \(\Omega\) is an (arbitrary) bounded domain in \(\mathbb{C}^n\). The squeezing function of \(\Omega\) is defined as follows. Let \(p\in\Omega\) and consider the set of holomorphic embeddings in the unit ball \(f:\Omega\to\mathbb{B}^n\) satisfying \(f(p)=0\). For each such \(f\) we define the real number \[ S_\Omega(p,f)=\sup\left\{ r>0: r\mathbb{B}^n\subset f(\Omega)\right\}. \] The squeezing function is the map \(S_{\Omega}:\Omega\to \mathbb{R}\) defined as \[ S_{\Omega}(p)=\sup_f\{S_\Omega(p,f)\} \] where the supremum is taken over all such embeddings. Squeezing functions are invariant under biholomorphic maps and have been the object of study as a means of understanding the geometric and analytic structures of bounded domains. It is well known that functions which are naturally defined on pseudoconvex domains often have plurisubharmonic properties and an open question has been whether it is always true that the squeezing function of a strictly pseudoconvex domain with smooth boundary is plurisubharmonic. In this paper the authors give a negative answer to this question by constructing a bounded strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic. The methods are based on explicit estimates involving the Kobayashi and Caratheodory metrics and the constructed squeezing function that is shown not to satisfy the maximum principle and thus cannot be plurisubharmonic.