High-entropy materials have attracted considerable interest due to their unique, improved properties and large configurational entropy. Out of these, high-entropy ceramics (HECs) are of particular interest since the independent solubility of cations and anions results in increased configurational entropy. However, most HEC research considers only a single element occupying the anion sublattice, which limits the maximum attainable configurational entropy. Here, we expand our previous work on high-entropy borocarbides where both boron and carbon occupy the anion sublattice. By applying an ab initio based screening procedure, we identify six elements Li, Ti, V, Zr, Nb and Hf suitable for forming high-entropy borocarbides. With these elements, we propose six novel HEC compositions, and by computing their entropy forming ability, we identify that three are likely to form single-phase during synthesis. Material properties and lattice distortions for all proposed compositions are studied using density functional theory calculations with special quasirandom structures. The directional lattice distortions, a concept we introduce in this work, show that lattice distortions have an elemental and directional preference for certain HEC compositions. We also show that the novel inclusion of Li improves the mechanical properties of the proposed HECs, the details of which are studied using the electron localization function.