Z-Axis doesn't work only during printing
I would check the gcode you're generating to make sure it's not full of &quot;bad&quot; z-commands.
Further, I'd take any gcode file you have and manually edit it, leaving in all the initializations, heatings, zeroing, etc.,  and then delete everything except some z-motion commands.  Run that as a test case to see what happens.
I  think it's extremely unlikely that the gantry has enough mass to force the z-screws to counter-rotate (lowering the gantry), and you'd see that with power off in any case.
edit
As Greenonline discovered, a bad choice of lead screw might cause the behavior you're seeing.  It would be a bit scary if a stock kit such as the one you bought provided high-pitch Z-screws.  Better check that out!