  The Casimir force due to {\it thermal} fluctuations (or pseudo-Casimir force)
was previously calculated for the perfect Bose gas in the slab geometry for
various boundary conditions. The Casimir pressure due to {\it quantum}
fluctuations in a weakly-interacting dilute Bose-Einstein condensate (BEC)
confined to a parallel plate geometry was recently calculated for Dirichlet
boundary conditions. In this paper we calculate the Casimir energy and pressure
due to quantum fluctuations in a zero-temperature homogeneous
weakly-interacting dilute BEC confined to a parallel plate geometry with
periodic boundary conditions and include higher-order corrections which we
refer to as Bogoliubov corrections. The leading order term is identified as the
Casimir energy of a massless scalar field moving with wave velocity equal to
the speed of sound in the BEC. We then obtain the leading order Casimir
pressure in a general three-dimensional rectangular cavity of arbitrary lengths
and obtain the finite-size correction to the parallel plate scenario.
