Spectroscopic observations obtained with the VLT of one planetary nebula (PN) in Sextans A and of five PNe in Sextans A and of several H ii regions in these two dwarf irregular galaxies are presented. The extended spectral coverage, from 320.0 to 1000.0 nm, and the large telescope aperture allowed us to detect a number of emission lines, covering more than one ionization stage for several elements (He, O, S, Ar). The electron temperature diagnostic O iii line at 436.3 nm was measured in all six PNe and in several H ii regions allowing for an accurate determination of the ionic and total chemical abundances by means of the Ionization Correction Factors method. For the time being, these PNe are the farthest ones where such a direct measurement of the electron temperature is obtained. In addition, all PNe and H ii regions were also modelled using the photoionization code CLOUDY (Ferland et al. 1998, PASP, 110, 761). The physico-chemical properties of PNe and H ii regions are presented and discussed. A small dispersion in the oxygen abundance of H ii regions was found in both galaxies: 12 + log (O/H)=7.6± 0.2 in Sextans A, and 7.8± 0.2 in Sextans B. For the five PNe of Sextans B, we find that 12 + log (O/H)=8.0± 0.3, with a mean abundance consistent with that of H ii regions. The only PN known in Sextans A appears to have been produced by a quite massive progenitor, and has a significant nitrogen overabundance. In addition, its oxygen abundance is 0.4 dex larger than the mean abundance of H ii regions, possibly indicating an efficient third dredge-up for massive, low-metallicity PN progenitors. The metal enrichment of both galaxies is analyzed using these new data.