  Using deep near-infrared Keck/MOSFIRE observations, we analyze the
rest-optical spectra of eight star-forming galaxies in the COSMOS and GOODS-N
fields. We reach integration times of $\sim$10 hours in the deepest bands,
pushing the limits on current ground-based observational capabilities. The
targets fall into two redshift bins -- 5 galaxies at $z \sim 1.7$ and 3 at $z
\sim 2.5$ -- and were selected as likely to yield significant auroral-line
detections. Even with long integration times, detection of the auroral lines
remains challenging. We stack the spectra together into subsets based on
redshift, improving the signal-to-noise ratio on the [O III] $\lambda 4364$
auroral emission line and, in turn, enabling a direct measurement of the oxygen
abundance for each stack. We compare these measurements to commonly-employed
strong-line ratios alongside measurements from the literature. We find that the
stacks fall within the distribution of $z>1$ literature measurements, but a
larger sample size is needed to robustly constrain the relationships between
strong-line ratios and oxygen abundance at high redshift. We additionally
report detections of [O I] $\lambda6302$ for nine individual galaxies and
composite spectra of 21 targets in the MOSFIRE pointings. We plot their line
ratios on the [O III] $\lambda 5008$/H$\beta$ vs. [O I] $\lambda
6302$/H$\alpha$ diagnostic BPT diagram, comparing our targets to local galaxies
and H II regions. We find that the [O I]/H$\alpha$ ratios in our sample of
galaxies are consistent with being produced in gas ionized by $\alpha$-enhanced
massive stars, as has been previously inferred for rapidly-forming galaxies at
early cosmic times.
