  We calculate the structure and neutron content of neutrino-heated MHD winds
driven from the surface of newly-formed magnetars (``proto-magnetars'') and
from the midplane of hyper-accreting disks, two of the possible central engines
for gamma-ray bursts (GRBs) and hyper-energetic supernovae (SNe). Both the
surface of proto-magnetars and the midplane of neutrino-cooled accretion flows
(NDAFs) are electron degenerate and neutron-rich (neutron-to-proton ratio n/p
>> 1). If this substantial free neutron excess is preserved to large radii in
ultra-relativistic outflows, several important observational consequences may
result. Weak interaction processes, however, can drive n/p to ~1 in the
nondegenerate regions that obtain just above the surfaces of NDAFs and
proto-magnetars. Our calculations show that mildly relativistic neutron-rich
outflows from NDAFs are possible in the presence of a strong poloidal magnetic
field. However, we find that neutron-rich winds possess a minimum mass-loss
rate that likely precludes simultaneously neutron-rich and ultra-relativistic
(Lorentz factor > 100) NDAF winds accompanying a substantial accretion power.
In contrast, proto-magnetars are capable of producing neutron-rich
long-duration GRB outflows ~10-30 seconds following core bounce for
sub-millisecond rotation periods; such outflows would, however, accompany only
extremely energetic events, in which the GRB + SN energy budget exceeds ~ 4e52
ergs. Neutron-rich highly relativistic outflows may also be produced during
some short-duration GRBs by geometrically thick accretion disks formed from
compact object mergers. The implications for r-process nucleosynthesis, optical
transients due to non-relativistic neutron-rich winds, and Nickel production in
proto-magnetar and NDAF winds are also briefly discussed.
