Increasing evidence suggests that He II proximity profiles in the quasar spectra at z ~ 3 - 4 are sensitive probes of quasar ages. But the development of their H I counterparts is difficult to trace and remains poorly constrained. We compare the UV spectra of 15 He II quasars with their high-resolution optical counterparts and find a significant correlation between the sizes of He II and H I proximity zones. The luminous quasar HE2347-4342 displays a null proximity zone in both He II and H I, suggesting that it is extremely young (age <0.2 Myr). Three other quasars also display small proximity zones for He II and H I. There is no evidence that a H I ionization zone expands considerably faster than its He II counterpart. The results suggest that the expansion of quasar ionizing fronts may be noticeably slower than the speed of light, and raise the possibility of distinguishing young and old quasars from the sizes of their H I proximity zones.