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We present results of the first search for like-sign dilepton (e±e±, e± μ±, μ± μ±) events associated with multijets and large missing energy using 106 pb-1 of data in pp̅ collisions at √s= 1.8 TeV collected during 1992–1995 by the CDF experiment. Finding no events that pass our selection, we examine pair production of gluinos (g̃) and squarks (q̃) in a constrained framework of the minimal supersymmetric standard model. At tanβ =2 and μ = -800 GeV/c2, we set 95% confidence level limits of Mg̃ > 221 GeV/c2 for Mg̃ –̴ Mq̃, and Mg̃ . 168 GeV/c2for Mq̃>>Mg̃ >, both with small variation as a function of μ.