  We present dark-matter minihalo models for the Ultra-Compact High Velocity HI
Clouds (UCHVCs) recently discovered in the 21 cm ALFALFA survey. We assume
gravitational confinement of 10^4 K HI gas by flat-cored dark-matter subhalos
within the Local Group. We show that for flat cores, typical (median)
tidally-stripped cosmological subhalos at redshift z=0 have dark-matter masses
of ~10^7 M_{sun} within the central 300 pc (independent of total halo mass),
consistent with the "Strigari mass scale" observed in low-luminosity dwarf
galaxies. Flat-cored subhalos also resolve the mass-discrepancy between
simulated and observed satellites around the Milky Way. For the UCHVCs we
calculate the photoionization-limited hydrostatic gas profiles for any
distance-dependent total observed HI mass and predict the associated
(projected) HI half-mass radii, assuming the clouds are embedded in distant (d
> 300 kpc) and unstripped subhalos. For a typical UCHVC (0.9 Jy km/s) we
predict physical HI half-mass radii of 0.18 to 0.35 kpc (or angular sizes of
0.6 to 2.1 arcmin) for distances ranging from 300 kpc to 2 Mpc. As a
consistency check we model the gas rich dwarf galaxy Leo T, for which there is
a well-resolved HI column density profile and a known distance (420 kpc). For
Leo T we find that a subhalo with M_{300} ~ 8 (+/-0.2) 10^6 M_{sun} best fits
the observed HI profile. We derive an upper limit of P_{HIM} < 150 K/cm^3 for
the pressure of any enveloping hot IGM gas at the distance of Leo T. Our
analysis suggests that some of the UCHVCs may in fact constitute a population
of 21-cm-selected but optically-faint dwarf galaxies in the Local Group.
