  A square-contact representation of a planar graph $G=(V,E)$ maps vertices in
$V$ to interior-disjoint axis-aligned squares in the plane and edges in $E$ to
adjacencies between the sides of the corresponding squares. In this paper, we
study proper square-contact representations of planar graphs, in which any two
squares are either disjoint or share infinitely many points.
  We characterize the partial $2$-trees and the triconnected cycle-trees
allowing for such representations. For partial $2$-trees our characterization
uses a simple forbidden subgraph whose structure forces a separating triangle
in any embedding. For the triconnected cycle-trees, a subclass of the
triconnected simply-nested graphs, we use a new structural decomposition for
the graphs in this family, which may be of independent interest. Finally, we
study square-contact representations of general triconnected simply-nested
graphs with respect to their outerplanarity index.
