The proton (ionized hydrogen) and the electron are both spin-$\frac 1 2$, thus fermions. From the rules for adding spins, the total spin of atomic hydrogen is either $0$ or $1$, and therefore atomic hydrogen is a boson.
The essential ingredient in describing ionization equilibrium is the chemical potentials for the various species. The chemical potential depends on the statistics: Boltzmann, Fermi-Dirac, or Bose-Einstein. Taking the chemical potentials to be those of ideal Boltzmann gases, we get the result called the Saha equation.
Now, the Fermi temperature for electrons at the densities in the chromosphere is at least a factor $\approx 100$ lower than the thermodynamic temperature given, which means it doesn't matter if we use Fermi-Dirac or Boltzmann statistics. The Fermi temperature is inversely proportional to the mass, so for protons and atomic hydrogen, using Boltzmann statistics is a forteriori justified. Thus the Saha equation is applicable.
There certainly are plasma regimes where electron degeneracy can be important, i.e. $T \lesssim T_F$, but then the density is something like six orders of magnitude higher than in the chromosphere.