  Persistent excesses in the spectra of gamma rays from the galactic center and
cosmic ray antiprotons can be explained by dark matter of mass $50\sim 65$\,GeV
annihilating into $b$ quarks, but this is typically hard to reconcile with
direct detection constraints. We resolve this tension using a simple class of
models, where dark matter is a pseudo-Nambu-Goldstone boson, having naturally
momentum-suppressed couplings to nuclei. Exploring the parameter space of the
model, we find that it can explain the cosmic ray anomalies while remaining
compatible with constraints from the relic abundance and annihilation in dwarf
spheroidal galaxies. In certain regions of parameter space, the Higgs-dark
matter coupling can help stabilize the Higgs potential up to the Planck scale.
The scalar partner of the dark matter is an extra Higgs boson, that can explain
a tentative diphoton excess observed by CMS, and an excess of $b\bar b$ signal
from LEP, if its mass is $\sim 96$\,GeV and the model is extended to include a
heavy scalar quark. This extended model predicts a monochromatic gamma-ray line
near 64 GeV, at a level close to current experimental sensitivity, from dark
matter annihilations in the galaxy.
