  We study neutral and charged Higgs boson production in association with stop
and sbottom squarks at the Large Hadron Collider, within the Supergravity
inspired Minimal Supersymmetric Standard Model. The phenomenological relevance
of such reactions is twofold. Firstly, they constitute a novel production
mechanism of Higgs particles, either through a decay of a heavier (anti)squark
into a lighter one or via a Higgs bremsstrahlung process. Secondly, their
production rates are extremely sensitive to the values assumed by the five
input parameters of the model, this possibly allowing one to put stringent
constraints on the latter. After an exhaustive scan of the parameter space, we
find that the majority of such processes could be detectable at high
luminosity, provided tan(beta) is large, tan(beta) > 30 (except in the case of
\tilde{t}_1\tilde{t}_1^* h and \tilde{t}_1\tilde{t}_2^* h final states, whose
detection is also possible for smaller values), that the universal soft
Supersymmetric breaking masses are in the ranges M_0 < 500 GeV and M_{1/2} <
220 GeV, and that the trilinear couplings are negative, A_0 < 0. We also point
out some sizable decay signatures and discuss their Standard Model (SM)
backgrounds. Finally, we derive compact analytical formulae of the
corresponding scattering matrix elements.
