Squaring a square and discrete Ricci flow

What does "tangent" mean? If two squares touch only at a vertex, are they tangent? According to the diagram provided, tangent seems to mean that two squares should meet along some (nontrivial) segment of their edges (for instance 23 and 19 meet at a vertex, even though they do not have an edge in common).

If meeting only at a vertex is not counted as being tangent, then the edge graph of a tetrahedron is a counterexample to the proposed theorem (it is not possible to tile a square into 4 squares with each square touching all other squares along a segment of an edge).

I don't know about having one vertex per square, but there is a similar very interesting construction with edges at squares. It does not answer your question but it will still surely interest you.

Specifically, given a rectangle of height $a$ and width $b$ tiled by squares, you can put a vertex on each horizontal line segment and one edge per square connecting its top and bottom sides. Now, label all edges by the side lenght of the corresponding square and orient them from top to bottom. By seeing the labels as both currents and voltages, note that the resulting graph satisfy Kirchoff two laws, except at the top and bottom vertices. This is fixed by adding an edge from the bottom vertex to the top one with current $b$ and voltage $-a$: we then obtain a $3$-connected planar graph satisfying Kirchoff laws everywhere.

Conversely, given a $3$-connected planar graph with a specified edge $e$, if we impose some voltage $V_1$ on $e$ and resistance $1$ on all other edges, we can solve for all the currents and voltages with Kirchoff laws and this will give us the graph of a rectangle dissection into squares. If the equivalent resistance happen to be $1$, this will mean $a=b$, i.e. the big rectangle is a square.

For more details, this idea is described in chapter 11 of Richard Stanley's Algebraic combinatorics.