We discuss the spectral analysis of a sample of 63 Active Galactic Nuclei (AGN) detected above a limiting flux of $S(8-24$ keV$)=7times10^{-14}$ 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.
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