The 2007–2009 solar minimum was the longest of the space age. We present the first of two companion papers on Chandra and XMM‐Newton X‐ray campaigns of Jupiter through February–March 2007. We find that low solar X‐ray flux during solar minimum causes Jupiter's equatorial regions to be exceptionally X‐ray dim (0.21 GW at minimum; 0.76 GW at maximum). While the Jovian equatorial emission varies with solar cycle, the aurorae have comparably bright intervals at solar minimum and maximum. We apply atomic charge exchange models to auroral spectra and find that iogenic plasma of sulphur and oxygen ions provides excellent fits for XMM‐Newton observations. The fitted spectral S:O ratios of 0.4–1.3 are in good agreement with in situ magnetospheric S:O measurements of 0.3–1.5, suggesting that the ions that produce Jupiter's X‐ray aurora predominantly originate inside the magnetosphere. The aurorae were particularly bright on 24–25 February and 8–9 March, but these two observations exhibit very different spatial, spectral, and temporal behavior; 24–25 February was the only observation in this campaign with significant hard X‐ray bremsstrahlung from precipitating electrons, suggesting this may be rare. For 8–9 March, a bremsstrahlung component was absent, but bright oxygen O⁶⁺ lines and best‐fit models containing carbon, point to contributions from solar wind ions. This contribution is absent in the other observations. Comparing simultaneous Chandra ACIS and XMM‐Newton EPIC spectra showed that ACIS systematically underreported 0.45‐ to 0.6‐keV Jovian emission, suggesting quenching may be less important for Jupiter's atmosphere than previously thought. We therefore recommend XMM‐Newton for spectral analyses and quantifying opacity/quenching effects.