Why are hydrogen, helium and neon known as quantum gases in the mid-20th-century chemical literature?
The usage you have found is at odds with the modern understanding of the term, which (as explained in the existing answer) tends to revolve around low-temperature behaviour, and can include all sorts of gases (say, all the way up to rubidium).
The passage you've quoted seems to be looking at different behaviour, and its meaning becomes clearer in the related paper
Vapor-Liquid Equilibria at High Pressures. Vapor-Phase Fugacity Coefficients in Nonpolar and Quantum-Gas Mixtures P. L. Chueh, and J. M. Prausnitz. Ind. Eng. Chem. Fundamen. 6, 492 (1967),
available as a pdf here, which makes the claim much more clear:
Quantum Gases
The configurational properties of low-molecular-weight gases (hydrogen, helium, neon) are described by quantum, rather than classical, statistical mechanics.
(The rest of that passage looks eerily similar to the one in your textbook. Is the Chueh & Prausnitz paper the reference 18 cited in your book? If it isn't, there's some pretty flagrant behaviour there.)
Basically, what they're claiming is that if you're studying the dynamics of a gas molecule leaving the liquid phase and into more open space, then classical mechanics is a good approximation so long as the molecule is massive enough, and that this approximation works well for all but the very lightest of molecules.
That's where your listing comes in: H$_2$, He and Ne are the lightest possible constituents of reasonable gases, as most everything in between will coalesce into diatomics that are heavier than neon. Presumably the claim goes that by the time you get to N$_2$ at mass 14 then the quantum mechanical effects become effectively negligible.
(And there are, of course, unreasonable gases ─ HF in particular, but also potentially Li$_2$ and Be$_2$ ─ which lie below that mass-$10$ cutoff, so presumably the fugacity calculations would need to be repeated for them, but I don't think that studying the equilibrium gas and liquid fractions of hydrofluoric acid as a function of temperature is a particularly appealing experiment.)
At high temperature, all of the elements you said will closely follow the behavior of ideal gas.
Those gases reach quantum degeneracy when temperature becomes cold enough such that the thermal de Broglie wavelength (inversely proportional to standard deviation in momentum - as temperature goes down, momentum spread decreases) starts to become comparable to interparticle spacing (see the Wikipedia article on Thermal de Broglie wavelength).
Another way to say the same thing is that the phase space density of the gas starts to approximate unity.
At this point, the quantum statistics of particles become important, and the quantum degenerate gas can be classified as Bose-Einstein condensate or Fermi degenerate gas, depending on whether the element is boson or fermion. I think all of the elements you listed only have bosonic isotopes.
Interacting Bose-Einstein condensates display superfluid behavior. Look up any popular article on BEC.
Now some miscellaneous points:
The first hydrogen BEC was created at MIT by Dan Kleppner and Tom Greytak. Most elements form solid at low temperature, but hydrogen stays gaseous. Actually you can make BEC with other elements (e.g. alkali), but they are metastable.
helium is special since it stays as fluid at low temperature. You need high pressure to make it solidify at low temperature. Superfluid helium is an example of strongly interacting superfluid, whereas other quantum degenerate gases are typically weakly interacting, unless you modify the interparticle scattering behavior using external fields.