Let $T$ be a tree with vertex set $\{1, \ldots, n\}$ such that each edge is assigned a nonzero weight. The squared distance matrix of $T,$ denoted by $��,$ is the $n \times n$ matrix with $(i,j)$-element $d(i,j)^2,$ where $d(i,j)$ is the sum of the weights of the edges on the $(ij)$-path. We obtain a formula for the determinant of $��.$ A formula for $��^{-1}$ is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.