  We discuss the spectral analysis of a sample of 63 Active Galactic Nuclei
(AGN) detected above a limiting flux of $S(8-24$ keV$)=7\times10^{-14}$
erg/s/cm$^2$ in the multi-tiered NuSTAR Extragalactic Survey program. The
sources span a redshift range z=0-2.1 (median $\langle$z$\rangle=$0.58). The
spectral analysis is performed over the broad 0.5-24 keV energy range,
combining NuSTAR with Chandra and/or XMM-Newton data and employing empirical
and physically motivated models. This constitutes the largest sample of AGN
selected at $ >10$ keV to be homogeneously spectrally analyzed at these flux
levels. We study the distribution of spectral parameters such as photon index,
column density ($N_{\rm H}$), reflection parameter ($R$) and 10-40 keV
luminosity ($L_{X}$). Heavily obscured ($\log [N_{\rm H}/cm^{-2}]\ge23$) and
Compton Thick (CT; $\log [N_{\rm H}/cm^{-2}]\ge24$) AGN constitute $\sim$25%
(15-17 sources) and $\sim$2-3% ( 1-2 sources) of the sample, respectively. The
observed $N_{\rm H}$ distribution fairly agrees with predictions of Cosmic
X-ray Background population synthesis models (CXBPSM). We estimate the
intrinsic fraction of AGN as a function of $N_{H}$, accounting for the bias
against obscured AGN in a flux-selected sample. The fraction of CT AGN relative
to $\log [N_{\rm H}/cm^{-2}]=20-24$ AGN is poorly constrainted, formally in the
range 2-56% (90% upper limit of 66%). We derived a fraction ($f_{abs}$) of
obscured AGN ($\log [N_{\rm H}/cm^{-2}]=22-24$) as a function of $L_{X}$ in
agreement with CXBPSM and previous z < 1 X-ray determinations. Furthermore
$f_{abs}$ at z=0.1-0.5 and $\log (L_{X}/erg$ $s^{-1})\approx43.6-44.3$ agrees
with observational measurements/trends obtained over larger redshift intervals.
We report a significant anti-correlation of $R$ with $L_{X}$ (confirmed by our
companion paper on stacked spectra) with considerable scatter around the median
$R$ values.
