If the mass of the charged Higgs boson of the minimal supersymmetric model (MSSM) is smaller than about 150 GeV, the branching fraction for heavy gluinos and left-handed squarks to cascade decay to the heavier Higgs bosons H h , H p , and H* of the MSSM may be as large as 50-60% for values of supersymmetric parameters consistent with experimental data. These decays, which have been assumed to be kinematically inaccessible in earlier analyses, can potentially lead to significant modification of the cross section for missing ET ( B T ) , same-sign dilepton, and multilepton events from squarks and gluinos. We find that the q T , same-sign dilepton, and trilepton production rates are relatively insensitive to low values of m~+ , so that these remain viable signals for the identification of gluinos and squarks at he Supercoliducting Super Collider or the CERN Large Hadron Collider. Finally, our exploratory study shows that substantial production of Hh and H , (H*) in gluino and squark pair events can lead to qT events with anomalously large numbers of b quarks ( r leptons). There also exist small regions of parameter space where the Hl and Hh may be identifiable via the presence of y y in qT events.