Why can't electrons be transferred using hydrogen ions?
Firstly, $\ce{H+}$ would be called a hydrogen ion, and $\ce{H-}$ would be called a hydride ion. It would take a lot of energy to get three electrons onto a hydrogen atom, and considering that we are talking about a biological system, this just won't happen.

$\ce{H+}$ would not be employed in the transfer of electrons because it is a lone proton, and has no electrons to give. Whatever electron the hydrogen atom originally had was left behind, and the proton now exists on its own. This, I believe, should explain your homework question.

To answer your question, $\ce{H-}$ ions are common reducing agents in non-biological organic reactions, in the forms of substances like $\ce{NaBH4}$ and $\ce{LiAlH4}$, but these are extremely unstable in aqueous solution. 



Instead, cells uses NAD+ to facilitate the transfer of electrons from biomolecules, like those used in cellular respiration, in the form of $\ce{H-}$. During the transfer, one electron from the $\ce{H-}$ is transferred to a nitrogen, and the other electron along with the proton forms a bond to a carbon directly across the ring from the nitrogen, converting NAD+ to NADH, a reaction that is readily reversible.