Re: why is a square structure stronger than a rectangle one?

I don't think that the shape has much to do with what's happening.  What 
I think you are observing is that a shorter member is stronger than a 
longer member.

As you may have noticed, a structural member that is loaded in 
compression -- that is, by being pushed down on as opposed to being 
pulled -- will suddenly go "K" shaped. Engineers call this "buckling 
failure".  You can demonstrate this for yourself by making two stacks of 
Legos, one long and one short.  The longer one will go--Sprang!  sideways 
under less load than the shorter one (Do this very carefully!) 

The average amount of stress a column can support is described by Euler's 
Formula:  f = (pi)^2 x E x (r / l)^2, 

where E is a property of the material (steel, concrete, Legos, etc), r is 
a property of the shape and cross-sectional area of the column , and l is 
the length.  Notice that if E and r are kept the same, the strength of the 
column varies inversely as the square of the length, i.e. doubling the 
column's length makes it four times weaker.

When your geometric shapes hit the floor, they are subjected to sudden, 
transient compressive and bending loads.  A longer Lego member, like the 
long side of the rectangle, will be less capable of withstanding this kind 
of loading than a shorter member, so the rectangle fails more often than 
the square.

Now how can we prove this?  Try making a Lego square with all 4 sides as 
long as the long side of the rectangle.  I'm not a betting man,  but I'd 
wager that this square fails as often as the rectangle -- probably more 
often, since it has two more long sides.  While you're at it, try making a 
rectangle whose long side is only as long as the square you had 
originally.  It should fail less often.  

Of course, the weight of the squares and rectangles will be different, 
since there will be different numbers of pieces in them, and that will 
affect the impact loads.  So you should add weight to the smaller shapes 
until they weigh exactly the same as the biggest shape -- possibly by 
putting little dabs of modeling clay inside of a few of the Legos.  (You 
won't  need to use a fancy balance to get the weights right -- hang a 
ruler from a string, and hook the test shapes to the ends.)

I've used Legos in my structural engineering demonstrations.  What you are 
trying sounds really interesting.  Good Luck.