What type of bond(s) are present in the borohydride ion?
No, this is simply a bond between four hydrogen atoms ($\ce{H}$) and a boron anion ($\ce{B-}$). These bonds are identical to and indistinguishable from the other $\ce{B-H}$ bonds. They are all covalent bonds of length $1.21\:\mathrm{Å}$. 

Your idea of $\ce{H-}$ completing $\ce{BH3}$'s $2\mathrm{p}_z$ orbital isn't quite right either. Atom's individually have atomic orbitals, but when they form bonds these atomic orbitals combine to form molecular orbitals. This is picture more accurately demonstrates the bonding between $\ce{B-}$ and the $\ce{H}$ atoms.



[Photo Source]