  Strongly-interacting matter in the form of nuggets of nuclear-density
material are not currently excluded as dark matter candidates in the ten gram
to hundred kiloton mass range. A recent variation on quark nugget dark matter
models postulates that a first-order imbalance between matter and antimatter in
the quark-gluon plasma prior to hadron production in the early universe binds
up most of the dark matter into heavy (baryon number $B \sim 10^{25}$)
anti-quark nuggets in the current epoch, explaining both the dark matter
preponderance and the matter-antimatter asymmetry. Interactions of these
massive objects with normal matter in the Earth and Sun will lead to
annihilation and an associated neutrino flux in the $\sim 20-50$ MeV range. We
calculate these fluxes for anti-quark nuggets of sufficient number density to
account for the dark matter and find that current neutrino flux limits from
Super-Kamiokande provide stringent constraints on several possible scenarios
for such objects. Conventional anti-quark nuggets in the previously allowed
mass range cannot account for more than $\sim 1/5$ of the dark matter flux; if
they are in a color-superconducting phase, then their muon production during
matter annihilation must be suppressed by an order of magnitude below prior
estimates if they are to remain viable dark matter candidates.
