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 106pb−1 of data in pp¯ collisions at s√=1.8TeV 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=−800GeV/c2, we set 95% confidence level limits of Mg̃>221GeV/c2 for Mg̃≃Mq̃, and Mg̃>168GeV/c2 for Mq̃≫Mg̃, both with small variation as a function of μ.