Are photons inside the media massive? If yes, why there is no Meissner effect? We all know in vacuum travels with speed $c$, hence its rest mass has to be 0. In the media the light speed $v<c$. Then the photon renormalized by the medium (call it "quasi-photon" if you like), has to have a nonzero rest mass $m_0\neq0$, since otherwise its energy $\epsilon=m_0c^2/\sqrt{1-v^2/c^2}=0$.
However, we also know that inside a superconductor the photon gains mass by "eating a Goldstone boson". The consequence of a photon mass is that the electromagnetic field is screened and we have a Meissner effect.
Both arguments seem rather general. So my questions are
(a) does a photon gain mass in a regular medium?
(b) If the answer is yes, why there is no Meissner effect (such that no light goes through, which will invalidate the name "medium") in regular media, such as our atmosphere?
 A: Suppose light, i.e. a quantum superposition of free photons and excited matter states, is travelling in a medium with refractive index $n$. Then we can in principle boost to an inertial frame which is at rest relative to the superposition travelling in the medium: we see a stationary region of excitation as we ride alongside the glass, and the latter zips by at a speed $c/n$ relative to our new inertial frame. By definition, then, the rest mass of the superposition is the energy of this disturbance divided by $c^2$, a nonzero number. Reasoning along these lines, I calculate in this answer here the following rest mass for the photon/matter quasiparticle:
$$m_0 = \frac{E}{c^2}\sqrt{1-\frac{1}{n^2}}$$
where $E$ is the total energy of the disturbance as measured from the frame at rest relative to the medium and $n$ the refractive index. For windowpane superpositions ($n=1.5;\;\lambda = 500{\rm nm}$) the quasiparticle has a rest mass of about 3.6 millionths of the mass of an electron.
Why does this not lead to a Meissner / screening effect? I'll mostly defer to someone brighter than I am, but I think I can begin to give the beginnings of an answer. A fundamental photon with a rest mass fulfils a Lorentz covariant equation such as the Proca or Klein Gordon equation and thus we get the hallmark Yukawa exponential decay with distance in any EM potential, either static or retarded. But here, we have a medium, we don't have a fundamental particle but rather a superposition and so we expect any equation which models the medium simply as a refractive index not to be Lorentz covariant: in the frame at which the medium is at rest, we have an (usually) isotropic refractive index: when we boost to a frame where a particular photon / matter state superposition is at rest, our refractive index is highly anisotropic, being $n$ in a direction orthogonal to the motion and infinite (since the pulse stands still) in a direction along the motion. Experimentally, the equations of propagation for this superposition are Maxwell's equations with $c$ replaced by $c/n$ (through the corresponding alteration of the electric/ magnetic constants) in the frame at rest relative to the medium and not a covariant Proca equation.
A: First, thanks to all who provided answers and ideas above. They really helped clarify things greatly. I think I have come to a plausible answer below.
If one wants to consider a "quasi-photon", then the medium (consisting of ions, localized electrons etc) has to be treated as a part of the vacuum upon which you build your field theory. However, this medium (or "quasi-vacuum") is not Lorentz invariant, because, for example, if you boost your medium to a different frame its density changes and hence $n$ changes. Therefore, you are doomed to end up with a field theory for quasi-photons that is not Lorentz invariant. In such a non-relativistic field theory, traveling with speed $v<c$ does not necessarily mean the corresponding particle is massive, or in condensed matter terms, gapped. Indeed one has all kinds of excitations with gapless linear dispersion but with speed smaller than $c$. Phonon is one great example. I think @Rococo was the first who realized this in this post. In this sense, relativity does NOT apply to quasiparticles, including quasi-photon!
On the other hand, suppose a photon traveling in a medium with $n=1.5$ were massive, then this mass is of order of its own energy, which corresponds to a screening length of its own wavelength -- a light wave wouldn't be able to propagate for even one wave length! This is in gross contradiction with the name "medium". I'd like to thank @ WetSavannaAnimal aka Rod Vance for first doing a similar computation.
Since photon in medium do not have mass (neglecting screening effects), there is no Meissner effect.
A: Okay, I had all sorts of complicated thoughts about this but I think it is actually very simple: the photon in a medium doesn't have an effective mass, and Rod Vance's equation is actually therefore incorrect.
Call it a proof by contradiction: a nonzero mass for a photon in a medium would mean that there is some nonzero energy that a photon must have to propagate in that medium, but for an ideal dielectric this is not the case. Indeed, as Lubos pointed out in the original discussion about this, the energy of a photon doesn't change upon entering a medium, so one can send in photons of arbitrarily low energy and have them propagate. So there cannot be a photon mass. The dispersion relation in the dielectric must be perfectly linear, just scaled by the index of refraction away from the free space dispersion. Although the context is quite different, in spirit this is very much like the "massless" Dirac cone quasiparticles in graphene that propagate at some material-dependent speed slower than $c$.
A: I think the question is best answered by disentangling different concepts, such as massive, screening, and superconductivity. Any of these may characterize a media, but not necessarily in the same time.
Massive
Mass can be viewed as a property of a dispersion relation, which does not have a linear term when expanded in wave vector:
$$
\hbar\omega(\mathbf{k})=\hbar\omega(0) + \sum_\alpha\hbar\frac{\partial \omega(\mathbf{k})}{\partial k_\alpha}|_{\mathbf{k}=0}k_\alpha
+ 
\sum_{\alpha,\beta}\hbar\frac{\partial^2 \omega(\mathbf{k})}{\partial k_\alpha \partial k_\beta}|_{\mathbf{k}=0}k_\alpha k_\beta
=\Delta + \hbar v|\mathbf{k}| + \frac{\hbar^2\mathbf{k}^2}{2m}
$$
(assuming spherical symmetry.) If $v=0$, we have massive modes/excitations. If, in addition, $\Delta>0$, these modes are also gapped.
Thus, massive means that modes of certain frequency cannot exist in the media. This is not necessarily because they are damped - they might be simply unable to penetrate it (as, e.g., as a result of destructive interference, like in periodic structures.)
Plasma is an example of a media where photons acquire mass (see Photon effective mass in plasma). Note that by plasma we mean here not necessarily ionized gas, but also the electronic plasma in metals. The peculiar dispersion relation has to do with the existence of plasma oscillations in either ionized gas or metal (plasmons), which are strongly coupled to light.
There are however many media (e.g., many dielectrics) where the dispersion relation is linear, $\omega(\mathbf{k})=v|\mathbf{k}$
Screening
Screening is expulsion of electric field out of a material. This is due to the charges in the material, displaced by the electric field, creating the electric field in the opposite direction. Thus, electric field cannot penetrate far inside the metals, having high electron density. This does not mean that the electromagnetic waves become massive, but only that they decay when we go into depth of the material (skin effect). One could illustrate the difference between mass and damping in terms of Klein-Gordon/Wave equation as:
$$
\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2} + \frac{\gamma}{c}\frac{\partial\varphi}{\partial t} -\nabla^2\varphi + \frac{m^2c^2}{\hbar^2}\varphi=0$$
The mass term is the one does not containing derivative, whereas the damping term has first derivative in time - making the time part of the equation similar to the damped harmonic oscillator (this term may have more complex structure - it is generally not obtained direction from a Hamiltonian theory, but via considering interactions between the particles/waves and teh environment.)
As we see from the discussion above, normal metals generally would imply existence of plasma oscillation and screening, entailing both damping and emergence of mass. In ionized gas plasma, which is electrically neutral, screening/damping of the field is relatively weak, and the gapped nature of light is more important.
Meissner effect
Normal metals expel electric field, but not the magnetic field - at least when it is static, and not associated with electric field. When we induce a magnetic field in a metal, it obviously induces Faraday currents that resist the change of the field in the metal. However, due to the finite resistance of the normal metal, these currents eventually decay, while the magnetic field remains.
In a superconductor the Faraday currents are not damped, and compensate the magnetic field inside the superconductor for perpetuity. This is the Meissner effect.
A: Apparently the answer to whether a photon gain mass in media is "no", because a massive photon for v=2/3 c would lead to a Yukawa screening length of 1$\mu m$.
Another idea that came to me is that media does not generate mass for the photon, but rather directly changes the metric from (1,1,1,c) to (1,1,1,v). This way a photon remains massless, and there is no Meissner effect.
But details are still lacking...
