Can multi-TeV (top and other) squarks be natural in gauge mediation?

We investigate whether multi-TeV (1-3 TeV) squarks can be natural in models of gauge-mediated SUSY breaking. The idea is that for some boundary condition of the scalar (Higgs boson and top squark) masses, the Higgs boson (mass)(2), evaluated at the renormalization scale similar to O(100) GeV, is not very sensitive to (boundary values of) the scalar masses (this has been called "focusing" in recent literature). Then, the top squark masses can be multi-TeV without leading to fine-tuning in electroweak symmetry breaking. Minimal gauge mediation does not lead to this focusing (for all values of tan beta and the messenger scale): the (boundary value of) the Higgs boson mass is too small compared to the top squark masses. Also, in minimal gauge mediation, the gaugino masses are of the same order as the scalar masses so that multi-TeV scalars implies multi-TeV gauginos (especially the gluino) leading to fine-tuning. We discuss ideas to increase the Higgs boson mass relative to the top squark masses (so that focusing can be achieved) and also to suppress gaugino masses relative to scalar masses (or to modify the gaugino mass relations) in nonminimal models of gauge mediation-then multi-TeV (top and other) squarks can be natural. Specific models of gauge mediation which incorporate these ideas and thus have squarks (and, in some cases, the gluino) heavier than 1 TeV without resulting in fine-tuning are also studied and their collider signals are contrasted with those of other models which have multi-TeV squarks.