The equiatomic quaternary Heusler alloy CoFeCrAl is a candidate material for spin-gapless semiconductors (SGSs). However, to date, there have been no experimental attempts at fabricating a junction device. This paper reports a fully epitaxial (001)-oriented MgO barrier magnetic tunnel junction (MTJ) with CoFeCrAl electrodes grown on a Cr buffer. X-ray and electron diffraction measurements show that the (001) CoFeCrAl electrode films with atomically flat surfaces have a B2-ordered phase. The saturation magnetization is 380 emu/cm3, almost the same as the value given by the Slater–Pauling–like rule, and the maximum tunnel magnetoresistance ratios at 300 K and 10 K are 87% and 165%, respectively. Cross-sectional electron diffraction analysis shows that the MTJs have MgO interfaces with fewer dislocations. The temperature- and bias-voltage dependence of the transport measurements indicates magnon-induced inelastic electron tunneling overlapping with the coherent electron tunneling. X-ray magnetic circular dichroism (XMCD) measurements show a ferromagnetic arrangement of the Co and Fe magnetic moments of B2-ordered CoFeCrAl, in contrast to the ferrimagnetic arrangement predicted for the Y -ordered state possessing SGS characteristics. Ab-initio calculations taking account of the Cr-Fe swap disorder qualitatively explain the XMCD results. Finally, the effect of the Cr-Fe swap disorder on the ability for electronic states to allow coherent electron tunneling is discussed.