Why is a canard configuration more Structurally efficient for Rutan's Voyager?
I listened to John Roncz explanation more than a couple of times and figured the reason he was explaining. 

The first thing to understand is any aerodynamic surface placed in front of the CG is a DE-stabilizer. The aft wing is the stabilizing surface.

The high inertia of the fuel filled booms demand either a very torsionally stiff wing structure or else a very big stabilizing power characterized by higher tail volume. But torsional strength is very expensive in terms of wing weight because higher torsional loads imply a bigger secondary spar and thicker wing skins. 

Considering the fact that the Breguet range equation favors higher structural efficiency over the other parameters, It appears that Burt chose the second approach. This adds to the second benefit that wing and canard both subjects almost only to the bending loads. 

Quick number-crunching shows that with 10% static margin canard configuration leads to ~7 Times statically stable airplane than the conventionally configured one.



EDIT:
As of why GlobalFlyer with very similar mission requirements used conventional configuration could be tracked down to the following reasons. 


Due to the nature of the Voyager project, Burt's responsibility was only to design the airplane. The materialization of the design was on Dick and Jeana, he opted for the most simple structural solution with the minimal analysis he could come up with at the time. 
As per the previous analysis on tail volume comparison, it is clear that conventional configuration provides only minimal "P" gain for the longitudinal transfer function; meaning that dampening out a gust response, for example, could result in diverging oscillations or even a stall. At the time, I believe completely failsafe autopilot is not a feasible solution given the additional weight it put and/or risks it brings onboard. 
Also, Global Flyer demands even a higher fuel fraction due to fuel thirst jet engines, so losing the aerodynamic benefits of conventional configuration was not a feasible one. Instead, they ended up utilizing a fulltime autopilot.