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We compute the renormalized energy-momentum tensor of a free scalar field coupled to gravity on an (n + 1)-dimensional Einstein static universe (ESU), {\bb R}\times S^n , with arbitrary low energy effective operators (up to mass dimension n + 1). A generic class of regulators is used, together with the Abel-Plana formula, leading to a manifestly regulator independent result. The general structure of the divergences is analysed to show that all the gravitational couplings (not just the cosmological constant) are renormalized for an arbitrary regulator. Various commonly used methods (damping function, point-splitting, momentum cut-off and zeta function) are shown to, effectively, belong to the given class. The final results depend strongly on the parity of n. A detailed analytical and numerical analysis is performed for the behaviours of the renormalized energy density and a quantity s which determines if the strong energy condition holds for the 'quantum fluid'. We briefly discuss the quantum fluid backreaction problem, via the higher dimensional Friedmann and Raychaudhuri equations, observe that equilibrium radii exist and unveil the possibility of a 'Casimir stabilization of Einstein static universes'. [ABSTRACT FROM AUTHOR]