Electrically induced electron spin polarization is imaged in $n$-type ZnSe epilayers using Kerr rotation spectroscopy. Despite no evidence for an electrically induced internal magnetic field, current-induced in-plane spin polarization is observed with characteristic spin lifetimes that decrease with doping density. The spin Hall effect is also observed, indicated by an electrically induced out-of-plane spin polarization with opposite sign for spins accumulating on opposite edges of the sample. The spin Hall conductivity is estimated as $3\pm 1.5{\Omega }^{-1}{\mathrm{m}}^{-1}/|e|$ at 20 K, which is consistent with the extrinsic mechanism. Both the current-induced spin polarization and the spin Hall effect are observed at temperatures from 10 to 295 K.