Construct an example of graph $G$ without bridges, such that its square $G^2$ is non hamiltonian. Note: Since Fleischner's Theorem (the square of each 2-connected graph is Hamiltonian) and bridges are forbidden, the required graph should have at least one cut-vertex.
You can find an example of a bridgeless graph with cut points, whose square is not hamiltonian in this paper of Fleischner and Kronk. (I know the paper is in German, but the figure of the graph is on the first page.) Fleischner also mentions this example in his paper "The Square of Every Two-Connected Graph Is Hamiltonian".