We report results of an experiment conducted in May 1997 to measure CdZnTe background and background reduction schemes in space flight conditions similar to those of proposed hard X-ray astrophysics missions. A 1 cm^2 CdZnTe detector was placed adjacent to a thick BGO anticoincidence shield and flown piggybacked onto the EXITE2 scientific balloon payload. The planar shield was designed to veto background countsproduced by local gamma-ray production in passive material and neutron interactions in the detector. The CdZnTe and BGO were partially surrounded by a Pb-Sn-Cu shield to approximate the grammage of an X-ray collimator, although the field of view was still ~2 pi sr. At an altitude of 127000 feet we find a reduction in background by a factor of 6 at 100 keV. The non-vetoed background is 9 X 10^{-4} cts /cm^2-sec-keV at 100 keV, about a factor of 2 higher than that of the collimated (4.5 deg FWHM) EXITE2 phoswich detector. We compare our recorded spectrum with that expected from simulations using GEANT and find agreement within a factor of 2 between 30 and 300 keV. We also compare our results with those of previous experiments using passive lead and active NaI shields, and discuss possible active shielding schemes in future astronomy missions employing large arrays of CdZnTe detectors.