We discuss the Higgs scenario in the minimal supersymmetric extension of the Standard Model at e+e- linear colliders operating in the c.m. energy range between 300 and 500 GeV. Besides decays of the Higgs particles into ordinary fermions and cascade decays, we analyze also decays into gaugino/Higgsinos and in particular, neutral Higgs decays into the lightest supersymmetric particles which are invisible if R-parity is conserved. The cross sections for the various production channels of SUSY Higgs particles in e+e- collisions are discussed in detail. The lightest Higgs boson cannot escape detection, and in major parts of the MSSM parameter space all five Higgs particles can be observed.