  We extend our previous studies of the neutrino fluxes expected from
neutralino LSP annihilations inside the Sun to include variants of the minimal
supersymmetric extension of the Standard Model (MSSM) with squark, slepton and
gaugino masses constrained to be universal at the GUT scale, but allowing one
or two non-universal supersymmetry-breaking parameters contributing to the
Higgs masses (NUHM1,2). As in the constrained MSSM (CMSSM) with universal Higgs
masses, there are large regions of the NUHM parameter space where the LSP
density inside the Sun is not in equilibrium, so that the annihilation rate may
be far below the capture rate, and there are also large regions where the
capture rate is not dominated by spin-dependent LSP-proton scattering. The
spectra possible in the NUHM are qualitatively similar to those in the CMSSM.
We calculate neutrino-induced muon fluxes above a threshold energy of 10 GeV,
appropriate for the IceCube/DeepCore detector, for points where the NUHM yields
the correct cosmological relic density for representative choices of the NUHM
parameters. We find that the IceCube/DeepCore detector can probe regions of the
NUHM parameter space in addition to analogues of the focus-point strip and the
tip of the coannihilation strip familiar from the CMSSM. These include regions
with enhanced Higgsino-gaugino mixing in the LSP composition, that occurs where
neutralino mass eigenstates cross over. On the other hand, rapid-annihilation
funnel regions in general yield neutrino fluxes that are unobservably small.
