We present new X-ray observations of luminous heavily dust-reddened quasars (HRQs) selected from infrared sky surveys. HRQs appear to be a dominant population at high redshifts and the highest luminosities, and may be associated with a transitional "blowout" phase of black hole and galaxy co-evolution models... Despite this, their high-energy properties have been poorly known. We use the overall sample of $10$ objects with XMM-Newton coverage to study the high-energy properties of HRQs at $\left< L_{\rm bol} \right> = 10^{47.5}$ erg/s and $\left< z \right>= 2.5$. For the seven sources with strong X-ray detections, we perform spectral analyses. These find a median X-ray luminosity of $\left< L_{\rm 2-10\,keV} \right> = 10^{45.1}$ erg/s, comparable to the most powerful X-ray quasars known. The gas column densities are $N_{\rm H}=(1$-$8)\times 10^{22}$ cm$^{-2}$, in agreement with the amount of dust extinction observed. The dust to gas ratios are sub-Galactic, but are higher than found in local AGN. The intrinsic X-ray luminosities of HRQs are weak compared to the mid-infrared ($L_{\rm 6\mu m}$) and bolometric luminosities ($L_{\rm bol}$), in agreement with findings for other luminous quasar samples. For instance, the X-ray to bolometric corrections range from $\kappa_{\rm bol}\approx 50$-$3000$. The moderate absorption levels and accretion rates close to the Eddington limit ($\left< \lambda_{\rm Edd} \right>=1.06$) are in agreement with a quasar blowout phase. Indeed, we find that the HRQs lie in the forbidden region of the $N_{\rm H}$-$\lambda_{\rm Edd}$ plane, and therefore that radiation pressure feedback on the dusty interstellar medium may be driving a phase of blowout that has been ongoing for a few $10^{5}$ years. The wider properties, including [OIII] narrow-line region kinematics, broadly agree with this interpretation. (read more)