The Higgs potential and the mass spectrum of the $N=1$ supersymmetric extension of a recently proposed one family model based on the local gauge group ${\mathrm{SU}(3)}_{C}\ensuremath{\bigotimes}{\mathrm{SU}(3)}_{L}\ensuremath{\bigotimes}{U(1)}_{X},$ which is a subgroup of the electroweak-strong unification group ${E}_{6},$ is analyzed. In this model the slepton multiplets play the role of the Higgs scalars and no Higgsinos are needed, with the consequence that the sneutrino, the selectron, and six other sleptons play the role of the Goldstone bosons. We show how the $\ensuremath{\mu}$ problem is successfully addressed in the context of this model which also predicts the existence of a light $\mathrm{CP}$-odd scalar.