  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 and stop) masses, the Higgs (mass)$^2$, evaluated at the
renormalization scale $\sim O(100)$ GeV, is not very sensitive to (boundary
values of) the scalar masses (this has been called ``focussing'' in recent
literature). Then, the stop masses can be multi-TeV without leading to
fine-tuning in electroweak symmetry breaking. {\em Minimal} gauge mediation
does {\em not} lead to this focussing (for all values of $\tan \beta$ and the
messenger scale): the (boundary value of) the Higgs mass is too small compared
to the stop 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 gluino) leading to fine-tuning. We discuss ideas to {\em
increase} the Higgs mass relative to the stop masses (so that focussing can be
achieved) and also to {\em suppress} gaugino masses relative to scalar masses
(or to modify the gaugino mass relations) in {\em non-minimal} 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 a 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.
