  We compute for the first time QCD corrections to the rare decays B -> X_{d,s}
nu nubar, B_{d,s} -> l^+ l^-, K -> pi nu nubar and K_L -> mu^+ mu^-, where l=e
or mu, in the context of a supersymmetric extension of the Standard Model (SM)
with minimal flavour violation and new operators, in addition to those present
in the SM. Assuming that the gluino is heavy, we consider an effective theory
which consists of charged and neutral Higgs particles, charginos and squarks.
We evaluate the QCD corrections to box and Z^0-penguin diagrams with top-quark,
charged Higgs boson, chargino and squark exchanges, as well as to neutral Higgs
boson penguin diagrams. We provide a compendium of analytic formulae for the
Wilson coefficients, which are valid for arbitrary values of tan(beta) (the
ratio of the vacuum expectation values of the two Higgs fields) except for the
case of the neutral Higgs-boson contributions. These contributions have been
obtained at large tan(beta), which may compensate for the inevitable
suppression by the masses of the light leptons in decays based on the b -> s
(d) l^+ l^- transition. We investigate the dependence of the various branching
ratios on the renormalization scale mu, which is the main theoretical
uncertainty in the short-distance calculation. We find that the mu dependence
of the branching ratios is considerably reduced once the QCD corrections are
taken into account. The contributions of new operators are found to be dominant
at large tan(beta) in B_{d,s} -> mu^+ mu^- while they are subleading in B ->
X_{d,s} nu nubar and completely negligible in kaon decays.
