  We study the boundedness of the $H^{\infty}$ functional calculus for
differential operators acting in (L^{p}(\mathbb{R}^{n};\mathbb{C}^{N})). For
constant coefficients, we give simple conditions on the symbols implying such
boundedness. For non-constant coefficients, we extend our recent results for
the (L^p) theory of the Kato square root problem to the more general framework
of Hodge-Dirac operators with variable coefficients (\Pi_B) as treated in
(L^2(\mathbb{R}^{n};\mathbb{C}^{N})) by Axelsson, Keith, and McIntosh. We
obtain a characterization of the property that (\Pi_B) has a bounded
(H^{\infty}) functional calculus, in terms of randomized boundedness conditions
of its resolvent. This allows us to deduce stability under small perturbations
of this functional calculus.
