You have not been very specific about boundary conditions. Still, the constant mean curvature surfaces (they are not called minimal if the mean curvature is nonzero) of revolution in $\mathbb R^3$ are the circular cylinder and the Delaunay surfaces,
http://en.wikipedia.org/wiki/Constant-mean-curvature_surface 


Well, I am not done, but let me at least suggest that the free boundary problem you appear to be describing ought to have the surface meeting the planes orthogonally. Furthermore, I think that the actual minimum surface area for a prescribed volume will simply be a circular cylinder of soap film. It is easy enough to prove that the minimizing surface cannot have a pinched waist and meet the planes obtusely, as would the catenoid. More work to be done.  


Anyway, see https://mathoverflow.net/questions/50300/is-there-a-complete-classification-of-constant-mean-curvature-surfaces  for a beginning.