Do Maxwell's equation describe a single photon or an infinite number of photons?
Because photons do not interact, to very good approximation for frequencies lower than $m_e c^2 / h$ ($m_e$ = electron mass), the theory for one photon corresponds pretty well to the theory for an infinite number of them, modulo Bose-Einstein symmetry concerns. This is similar to most of the statistical theory of ideal gases being derivable from looking at the behavior of a single gas particle in kinetic theory.
Put another way, the single photon behavior $\leftrightarrow$ Maxwell's equations correspondence only holds if you look at the Fourier transform version of Maxwell's equations. The real space-time version of Maxwell's equations would require looking at a superposition of an infinite number of photons — one way to describe the taking an inverse Fourier transform.
If you want to think of it in terms of Feynman diagrams, classical electromagnetism is described by a subset of the tree-level diagrams, while quantum field theory requires both tree level and diagrams that have closed loops in them. It is the fact that the lowest mass particle photons can produce a closed loop by interacting with, the electron, that keeps photons from scattering off of each other.
In sum: they're both incorrect for not including frequency cutoff concerns (pair production), and they're both right if you take the high frequency cutoff as a given, depending on how you look at things.
Do Maxwell's equation describe a single photon or an infinite number of photons?
Both.
(i) The single photon wave function is a solution of the free Maxwell equations in vacuum, and any nonzero solution of the free Maxwell equations in vacuum is a possible single photon wave function.
(ii) The mode of a coherent state of (arbitrarily many) photons is a solution of the free Maxwell equations in vacuum, and any nonzero solution of the free Maxwell equations in vacuum is a possible mode of a coherent state of photons. (The zero solution corresponds to the coherent state of zero intensity, usually called the vacuum state.)
For full details see the entry ''What is a photon?'' in Chapter B2 of my theoretical physics FAQ.
The above follows easily from the pure electromagnetic sector of QED (i.e., no matter present). The full strength of QED is needed to properly describe interactions of photons with electrons and other charged matter on a microscopic level.
More generally, for any free bosonic field theory (relativistic or not) there is a 1-1 correspondence between modes of coherent states and state vectors (wave functions) of the 1-particle Hilbert space of the theory.
I think that the authors are sloppily combining two quite distinct ideas into a single sentence.
The first half of the sentence, "Maxwell's theory can be considered as the quantum theory of a single photon," refers to the fact that the single tree-level vertex of QED involves one photon interacting with an electron/positron. So as Sean Lake said, tree-level QED is linear in photon amplitudes, in the sense that photons cannot scatter off each other (because doing so requires the exchange of closed loops of virtual electrons), and in this tree-level classical limit the photon dynamics are described by Maxwell's equations.
In the second half of the sentence, "geometrical optics [can be considered] as the classical mechanics of this photon," when the authors say "photon" they're really referring to the much older concept of a corpuscle. In the high-frequency limit - more precisely, the limit where the wavelength of light is much smaller than the size of the objects that it scatters off of - Maxwell's equations reduce to the much simpler theory of geometrical or ray optics, in which light rays simply travel in straight lines and can be thought of as being transmitted by discrete classical-mechanical particles called "corpuscles." These particles are very different from photons, as they are completely classical and have no wavelike nature whatsoever.
So indeed a single photon at very high frequency is technically formally similar to a Newtonian corpuscle. But this isn't very useful, because if there's even one other photon (which, realistically, there will be), then the photons will exchange virtual electrons and scatter in a very non-classical way. The more natural way to get geometrical optics is to get a huge number of photons propagating in a semiclassical coherent state (so that as Sakurai says, you can ignore their discrete nature), with a wavelength that is small compared to the size of the scattering objects (so that you can ignore the classical wavelike behavior of Maxwell's equations) but large compared to the electron Compton wavelength (so that you can ignore QED scattering effects).