Actually you can argue that circular polarization is the "more natural" basis for a single photon. The photon carries one unit $\hbar$ of angular momentum, and circularly polarized light carries real angular momentum (an opportunity for me to mention one of my favorite experiments ever, using photon polarization to drive a pendulum).
A single photon that's linearly polarized has its spin in a superposition of states, equally likely to be found parallel or antiparallel to its momentum. As in your transformation from linear to polarized light, the relative phase of the spin components determines the direction of the linear polarization.
So it absolutely makes sense to talk about "a circularly polarized photon," and people do it all the time.
However I can't decide whether I agree with you about whether it makes sense to talk about "an unpolarized photon." Certainly you can make unpolarized light, and from that distribution of photons we can always draw individuals for which there is no way to predict their polarization in any basis. But any photon state that you could write down would have some polarization (most likely elliptical, with a randomly-oriented long axis direction). If it were the case that single photons are always coherently polarized with themselves, but ensembles of photons may average to zero polarization, there would be some interferometer trick you could use to distinguish between lone-photon and multiple-photon interference. I suspect that there isn't, but I'm not sure.