Is the Calibron 12 puzzle NP-hard?
It depends what you mean by this puzzle. If you mean specifically Calibron 12 then no, there is no growth factor (there are only ever 12 pieces) so the solution can be found in constant time (a very slow constant).   

If however you want to know about puzzles of this nature it is hard to determine. The problem could potentially be phrased as, Is there some arrangement of these $n$ objects that can completely fill this square. Now to me this does not seem likely to be polynomial-time in nature (I am speculating based on the permutations of arrangements, but I do not have a proof).

The bin packing comparison does not really work however since this has almost nothing to do with separating $n$ objects into  the smallest number of bins. This is a category of questions called tiling problems.

Here is a paper that deals with Rectangle tiling, which is a related problem.