A class of graphs $\mathcal{G}$ is $χ$-bounded if there exists a function $f$ such that $χ(G) \leq f(ω(G))$ for each graph $G \in \mathcal{G}$, where $χ(G)$ and $ω(G)$ are the chromatic and clique number of $G$, respectively. The square of a graph $G$, denoted as $G^2$, is the graph with the same vertex set as $G$ in which two vertices are adjacent when they are at a distance at most two in $G$. In this paper, we study the $χ$-boundedness of squares of bipartite graphs and its subclasses. Note that the class of squares of graphs, in general, admit a quadratic $χ$-binding function. Moreover there exist bipartite graphs $B$ for which $χ\left(B^2\right)$ is $Ω\left(\frac{\left(ω\left(B^2\right)\right)^2 }{\log ω\left(B^2\right)}\right)$. We first ask the following question: "What sub-classes of bipartite graphs have a linear $χ$-binding function?" We focus on the class of convex bipartite graphs and prove the following result: for any convex bipartite graph $G$, $χ\left(G^2\right) \leq \frac{3 ω\left(G^2\right)}{2}$. Our proof also yields a polynomial-time $3/2$-approximation algorithm for coloring squares of convex bipartite graphs. We then introduce a notion called "partite testable properties" for the squares of bipartite graphs. We say that a graph property $P$ is partite testable for the squares of bipartite graphs if for a bipartite graph $G=(A,B,E)$, whenever the induced subgraphs $G^2[A]$ and $G^2[B]$ satisfies the property $P$ then $G^2$ also satisfies the property $P$. Here, we discuss whether some of the well-known graph properties like perfectness, chordality, (anti-hole)-freeness, etc. are partite testable or not. As a consequence, we prove that the squares of biconvex bipartite graphs are perfect.