Many science and engineering applications use high-resolution unstructured hexahedral meshes to model solid 3D shapes for finite element simulations. These simulations frequently dump the mesh and associated fields to disk for subsequent analysis, which involves the transfer of huge volumes of data. To reduce requirements on disk space and bandwidth, we propose efficient schemes for lossless online compression of hexahedral mesh geometry and connectivity. Our approach is to use hash-based value predictors to transform the mesh connectivity list into a more compact byte-aligned stream of symbols that can then be efficiently compressed using conventional text compressors such as gzip. Our scheme is memory efficient, fast, and simple to implement, and yields 1-3 orders of magnitude reduction on a set of benchmark meshes. For geometry and field coding, we derive a set of local spectral predictors optimized for each possible configuration of previously encoded and thus available vertices within a hexahedron. Combined with lossless floating-point residual coding, this approach improves considerably upon prior predictive geometry coding schemes.