  We introduce a new class $\mathcal{G}$ of bipartite plane 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.
  For every graph $G\in \mathcal{G}$, we construct a proper square contact
representation. The key parameter of the recursive construction is the aspect
ratio of the rectangle bounded by the four outer squares. We show that this
aspect ratio may continuously vary in an interval $I_G$. The interval $I_G$
cannot be replaced by a fixed aspect ratio, however, as we show, the feasible
interval $I_G$ may be an arbitrarily small neighborhood of any positive real.
