Recent works have suggested that selection criteria based on mid-IR properties, i.e. extreme colors and bright flux levels, can be used to reveal a population of dust-enshrouded, extremely-luminous quasars at $z \sim 1$–2. However, the X-ray spectral properties of these intriguing objects still remain largely unexplored. We have performed an X-ray study of a large sample of bright mid-IR ($F_{24~\mu {\rm m}} > 1.3$ mJy) galaxies showing an extreme MIR/Optical flux ratio ($F_{24~\mu {\rm m}}$/$F_{\rm R}>2000$) in order to confirm the presence of a luminous active nucleus in these very red objects. Sampling of a large area is required to pick up objects at the highest luminosities given their low surface density. Accordingly, we have applied our selection criteria to an area of ~6 deg2 covered by XMM-Newton/Chandra observations within the ~50 deg2 SWIRE survey, resulting in a final sample of 44 objects. The vast majority of the source redshifts, both spectroscopic and photometric, are in the range $0.7 \la~z \la~2.5$. The X-ray coverage of the sample is highly inhomogeneous (from snap-shot 5 ks Chandra observations to medium-deep XMM-Newton exposures of 70 ks) and, consequently, a sizable fraction of them (≈43%) remains undetected in the 0.5–10 keV band. Using spectral or hardness information we were able to estimate the value of the absorbing column density in 23 sources. 95% of them are consistent with being obscured by neutral gas with an intrinsic column density of NH ≥ 1022 cm-2. Remarkably, we also find that ~55% of these sources can be classified as type 2 quasars on the basis of their absorption properties and X-ray luminosity. Moreover, most of the X-ray undetected sources show extreme mid-IR colors, consistent with being luminous AGN-powered objects, suggesting they might host heavily obscured (possibly Compton-thick) quasars in X-rays. This demonstrates that our selection criteria applied to a wide area survey is very efficient in finding a large number of type 2 quasars at $z \ga 1$. The existence of this class of very powerful, obscured quasars at high z could have important implications in the context of the formation and cosmological evolution of accreting supermassive black holes and their host galaxies.