We discuss the spectral analysis of a sample of 63 active galactic nuclei (AGN) detected above a limiting flux of in the multi-tiered NuSTAR extragalactic survey program. The sources span a redshift range (median ). 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 to be homogeneously spectrally analyzed at these flux levels. We study the distribution of spectral parameters such as photon index, column density ( ), reflection parameter ( ), and 10-40 keV luminosity ( ). Heavily obscured ( ) and Compton-thick (CT; ) AGN constitute ∼25% (15-17 sources) and ∼2-3% (1-2 sources) of the sample, respectively. The observed distribution agrees fairly well with predictions of cosmic X-ray background population-synthesis models (CXBPSM). We estimate the intrinsic fraction of AGN as a function of , accounting for the bias against obscured AGN in a flux-selected sample. The fraction of CT AGN relative to AGN is poorly constrained, formally in the range 2-56% (90% upper limit of 66%). We derived a fraction (fabs) of obscured AGN ( ) as a function of in agreement with CXBPSM and previous X-ray determinations. Furthermore, fabs at and agrees with observational measurements/trends obtained over larger redshift intervals. We report a significant anti-correlation of R with (confirmed by our companion paper on stacked spectra) with considerable scatter around the median R values.