How to keep Z-axis motor from slipping when powered down
The simple way to do this is to use a self-locking screw pitch. Pretty much any single-start thread using a sliding nut cannot be back-driven so the load will not fall. Normal 8x8 trapezoidal thread screws will easily back-drive because of the steep pitch. 

Likewise, a worm drive between the motor and Z stage will hold the load. You would want to switch from screws to belts for the main motion stage in that case though, to avoid having too much total gear reduction. 

Both of these solutions will limit your maximum Z speed, of course. But they're simple and reliable. Clutches and brakes add a lot of complexity and must be actuated somehow. Designers who want the load to stay suspended almost always simply use single-start screws.