Over the past 15 years, examples of exotic radio-quiet quasars with intrinsically weak or absent broad emission line regions (BELRs) have emerged from large-scale spectroscopic sky surveys. Here, we present spectroscopy of seven such weak emission line quasars (WLQs) at moderate redshifts (z=1.4-1.7) using the X-shooter spectrograph, which provides simultaneous optical and near-infrared spectroscopy covering the rest-frame ultraviolet through optical. These new observations effectively double the number of WLQs with spectroscopy in the optical rest-frame, and they allow us to compare the strengths of (weak) high-ionization emission lines (e.g., CIV) to low-ionization lines (e.g., MgII, Hb, Ha) in individual objects. We detect broad Hb and Ha emission in all objects, and these lines are generally toward the weaker end of the distribution expected for typical quasars (e.g., Hb has rest-frame equivalent widths ranging from 15-40 Ang.). However, these low-ionization lines are not exceptionally weak, as is the case for high-ionization lines in WLQs. The X-shooter spectra also display relatively strong optical FeII emission, Hb FWHM