We used VLT/XShooter to target a sample of nearby analogs of Lyman Break Galaxies (LBGs). These Lyman Break Analogs (LBAs) are similar to the LBGs in many of their physical properties. We determine electron temperatures using the weak [O III]λ4363 emission line, and determine the oxygen abundance (O/H) using the direct and strong line methods. We show that the direct and strong line abundances are consistent with established relations within ∼0.2 dex. The analogs have nitrogen-to-oxygen ratios (N/O) and ionization parameters (q) that are, on average, offset with respect to typical local galaxies but similar to galaxies at z ∼ 2 and other analogs. The N/O and q excesses correlate with the offsets observed in the strong line ratios, again similar to z ∼ 2. The star formation rate surface densities are consistent with the high electron density and ionization, indicating that the interstellar medium (ISM) pressure is set by feedback from the starbursts. For a given O/H, the apparent N/O excess arises due to the offset in O/H with respect to the local mass-metallicity relation. This can be explained by recent inflow of relatively metal-poor gas which lowers O/H while leaving N/O unchanged. The difficulties in determining even basic ISM parameters in these nearby analogs illustrates some of the challenges we face at much higher redshifts, where similar rest-frame optical diagnostics for large samples of galaxies can be accessed with JWST.