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.
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