They're states but not eigenstates. It's like the difference between RGB colour and greyscale. In both cases there are infinitely many possible colours, but in greyscale they're all mixtures of two "primary" colours (black and white) whereas in RGB they're mixture of four (black, red, green and blue).

In a qubit the infinitely many superposition states are all mixtures of just two eigenstates.

A superposition is indeed a state, comprised of linear combinations of the basis states.

Further, (and anyone, please correct me where I'm wrong), the eigenfunctions (which could actually be called eigenstates) of an operator ARE the basis set, as they are orthonormal. (Right?)

One can justify the name using the fact that the state space is 2-dimensional.