For a single photon, the only similar physically meaningful question is whether the circular polarization is left-handed or right-handed. Quantum mechanics may predict the probabilities of these two answers. An experiment, a measurement of L/R, produces one of these answers, too. After the measurement, the photon is either left-handed or right-handed circularly polarized.
If a photon is prepared in a general state, it has nonzero probabilities both for L and R. In such a "superposition", we may perhaps say that the single photon has no circular polarization. This statement means that we are uncertain which of the polarizations will be measured if it is measured. But when the circular polarization is measured, one always gets an answer, according to the result of the measurement.
Linear polarizations are the simplest nontrivial superpositions of L and R. The absolute value of both coefficients, $c_L$ and $c_R$, is the same while the relative phase encodes the axis on which the photon is polarized.
The paper quoted in the question is completely wrong. An example of a very wrong statement is that the linearly polarized photon moving in the $z^+$ direction carries $J_z=0\cdot\hbar$. In reality, a linearly polarized photon or any photon is certain not to have $J_z=0\cdot\hbar$. A linearly polarized photon has the 50% probability to be $J_z=+1\cdot\hbar$ and 50% to have $J_z=-1\cdot\hbar$. The expectation value $\langle J_z\rangle = 0$ but it's still true that the value $J_z=0\cdot\hbar$ is forbidden.
A different question is the polarization of an electromagnetic wave. For a wave, e.g. light, one may distinguish left-right and right-handed and $x$-linearly and $y$-linearly and elliptic polarizations of all kinds one may think of. In terms of photons, a macroscopic electromagnetic wave is the tensor product of many photons. If all these tensor factors are linearly (or circularly) polarized, then the wave may be said to be linearly (or circularly) polarized. Because the polarization of the whole wave requires some correlation in the state of individual photons, a wave may be measured not to be circularly polarized in either direction. But an individual photon is always circularly polarized in one of the directions when the answer to this question is measured.
The paper may present a proposed experiments which may be done but what is completely invalid is the author's interpretation of this experiment – even "possible interpretations" before the experiment is actually performed. The correct description by quantum mechanics isn't included among their candidate theories with which they want to describe the experiment.