We present the results of an extended computational study of the electric and magnetic properties connected to Cotton-Mouton birefringences, on the trifluoro- and trichloroborides in the gas phase. The electric dipole polarizabilities, magnetizabilities, quadrupole moments, and higher-order hypersusceptibilities--expressed as quadratic and cubic frequency-dependent response functions--are computed within Hartree-Fock, density-functional, and coupled-cluster response theories employing singly and doubly augmented correlation-consistent basis sets and London orbitals in the magnetic property calculations. The results, which illustrate the capability of time-dependent density-functional theory for electron-rich systems, are compared with available experimental data. Revised values of both experimentally derived quadrupole moment of BF3, 2.72 +/- 0.15 a.u., and magnetizability anisotropy of BCl3, -0.45 +/- 0.09 a.u., both obtained in birefringence experiments that neglect the effects of higher-order hypersusceptibilities, are presented. In the theoretical limit the traceless quadrupole moments of BF3 and BCl3 are determined to be 3.00 +/- 0.01 and 0.71 +/- 0.01 a.u., respectively.