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 nonuniversal 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 eigen-states cross over. On the other hand, rapid-annihilation funnel regions in general yield neutrino fluxes that are unobservably small.