We present Nuclear Spectroscopic Telescope Array (NuSTAR) hard X-ray observations of two X-ray weak broad absorption line (BAL) quasars, PG 1004+130 (radio loud) and PG 1700+518 (radio quiet). Many BAL quasars appear X-ray weak, probably due to absorption by the shielding gas between the nucleus and the accretion-disk wind. The two targets are among the optically brightest BAL quasars, yet they are known to be significantly X-ray weak at rest-frame 2-10 keV (16-120 times fainter than typical quasars). We would expect to obtain ≈ 400-600 hard X-ray (≳ 10 keV) photons with NuSTAR, provided that these photons are not significantly absorbed (NH ≲ 1024 cm-2). However, both BAL quasars are only detected in the softer NuSTAR bands (e.g., 4-20 keV) but not in its harder bands (e.g., 20-30 keV), suggesting that either the shielding gas is highly Compton-thick or the two targets are intrinsically X-ray weak. We constrain the column densities for both to be NH ≈ 7 × 1024 cm-2 if the weak hard X-ray emission is caused by obscuration from the shielding gas. We discuss a few possibilities for how PG 1004+130 could have Compton-thick shielding gas without strong Fe Kα line emission; dilution from jet-linked X-ray emission is one likely explanation. We also discuss the intrinsic X-ray weakness scenario based on a coronal-quenching model relevant to the shielding gas and disk wind of BAL quasars. Motivated by our NuSTAR results, we perform a Chandra stacking analysis with the Large Bright Quasar Survey BAL quasar sample and place statistical constraints upon the fraction of intrinsically X-ray weak BAL quasars; this fraction is likely 17%-40%.