Lattice-matched high crystal quality Zn1 - xMgxSeyTe1 - y quaternary alloys are grown on ZnTe substrates. For high crystalline and homogeneous alloy growth, the molecular beam pressure ratio is controlled as P-Se < P-Zn + P-Mg < P-Te and (P-Se + P-Te)/(P-Zn + P-Mg) >>1. The lattice-matching condition to ZnTe substrates is obtained as y = 0.63 x by both the correlated-function-expansion method and experiment. The growth temperature is optimized in terms of the X-ray rocking curve and photoluminescence line width. The observed narrow temperature window (300-320 degrees C) for high-quality crystal growth is explained in terms of the bond energy difference of the binary alloys, which compose ZnMgSeTe. Mg composition is determined by RHEED intensity oscillations of ZnTe and ZnMgTe, which shows good agreement with the values estimated from X-ray analysis and RHEED intensity oscillations of MgTe. The photoluminescence properties of ZnMgSeTe are characterized by strong excitonic luminescence intensity with a narrow line width (5.95 meV) and very weak deep level luminescence. The luminescence intensity of typical ZnMgSeTe is 3.5 times stronger than that of bulk ZnSe under the same experimental condition. The narrow luminescence line width and small Stokes shift of ZnTe/ZnMgSeTe quantum wells are indicative of the homogeneity of the quaternary alloy.