Rare earth equiatomic quaternary Heusler (EQH) compounds with chemical formula RXVZ (R = Yb, Lu; X = Fe, Co, Ni; Z = Al, Si) have recently attracted much attention since these materials are easily prepared and they also provide interesting properties for future spintronic applications. In this work, rare Earth-based EQH compounds in three types of structures are theoretically investigated through first-principles calculations based on density functional theory. We find that most of the studied rare Earth EQH compounds exhibit magnetic ground states including ferro-, antiferro-, and ferri-magnetic phases. Owing to the nearly closed shell f orbital in Lu and Yb, the spin magnetic moments mainly come from the 3d transition metal elements. In particular, in the type I structure, a large portion (7 out of 12) of EQH compounds are ferromagnetic half-metals (HMs) with integer magnetic moments ranging from 1 to 3 μB. In the type II structure, YbFeVAl is found to be a rare case of antiferro-magnetic (AFM) half-metal with zero total magnetic moments. Surprisingly, we also discover an unusual magnetic semiconductor LuCoVSi in the type III structure with a total spin magnetic moment of 3.0 μB and an indirect band gap of 0.2 eV. The structural and magnetic stabilities such as formation energy, magnetization energy as well as the mechanical stabilities such as the bulk, shear, and Young's moduli, and Poisson's, and Pugh's ratios of these EQH compounds are also investigated. Most of the studied compounds exhibit mechanical stability under the mechanical stability criteria and show elastic anisotropy. Our work provides guidelines for experimental researchers to synthesize useful materials in future spintronic applications.