We explore the implications of the t-b-tau (and b-tau) Yukawa coupling unification condition on the fundamental parameter space and sparticle spectroscopy in the minimal gauge mediated supersymmetry breaking model. We find that this scenario prefers values of the CP-odd Higgs mass m(A) greater than or similar to 1 TeV, with all colored sparticle masses above 3 TeV. These predictions will be hard to test at LHC13 but they may be testable at the high-energy LHC (HE-LHC) 33 TeV or a 100 TeV collider. Both the t-b-tau and the b-tau Yukawa coupling unification prefer a relatively light gravitino with mass less than or similar to 30 eV, which makes it a candidate hot dark matter particle. However, it cannot account for more than 15% of the observed dark matter density.