  We introduce a new class $\mathcal{G}$ of plane bipartite graphs and prove
that each graph in $\mathcal{G}$ admits a proper square contact representation.
A contact between two squares is \emph{proper} if they intersect in a line
segment of positive length. The class $\mathcal{G}$ is the family of
quadrangulations obtained from the 4-cycle $C_4$ by successively inserting a
single vertex or a 4-cycle of vertices into a face.
