  We study the zero point energy of massless scalar and vector fields subject
to spheroidal boundary conditions. For massless scalar fields and small
ellipticity the zero-point energy can be found using both zeta function and
Green's function methods. The result agrees with the conjecture that the zero
point energy for a boundary remains constant under small deformations of the
boundary that preserve volume (the boundary deformation conjecture), formulated
in the case of an elliptic-cylindrical boundary. In the case of massless vector
fields, an exact solution is not possible. We show that a zonal approximation
disagrees with the result of the boundary deformation conjecture. Applying our
results to the MIT bag model, we find that the zero point energy of the bag
should stabilize the bag against deformations from a spherical shape.
