I did as suggested and ran:

R600_HYPERZ=0 /usr/lib/misc/xscreensaver/antmaze

This worked more often, but I could make it crash when using different
screensavers. Using

R600_HYPERZ=0 /usr/bin/xscreensaver-demo

and switching between screensavers crashed as often as without R600_HYPERZ=0.
Also, the screensaver juggler3d crashed twice when exiting the window it ran
in.

R600_DEBUG=sb didn't help either.

I also confirmed that environment variables are working as they should via
setting GALLIUM_HUD.