The difference for superconductors is the following. Suppose you have a
superconducting ring in a magnetic field. From the Schrödinger equation
for a particle in an magnetic field, the phase change along an
arbitrary closed loop, C, is given by

\hbar c/Q Delta phi=\int_C d \vec{r}[\vec{A}(\vec{r)+ m c/Q^2
\vec{j}_{el}(\vec{r})/|\psi(\vec{r})|^2],

where \psi is the wave function of the particle with mass, m, and
electric charge, Q. \vec{A} is the vector potential describing the
external magnetic field.

According to Stoke's Law the first term on the rhs of this equation is
the magnetic flux.

Since the wave function has to be uniquely defined, this phase shift
must be an integer multiple of 4 pi.

There's no restriction for the magnetic flux, if you put your path
through normal-conducting material since there the electric current can
have an arbitrary value.

However, if you have superconducting material, the electric current has
to vanish strictly inside this material, and thus putting your
integration path along the superconducting ring in the above mentioned
situation, you have

\hbar c/Q \Delta phi=\Phi=\hbar c/Q n = n \Phi_0,

where n \in Z, i.e., the magnetic flux, Phi, through the superconducting
ring must be an integer multiple of the London-Flux Quantum,

Phi_0=\hbar c/Q

The experimental value is Q=-2 e, showing that the super current is
indeed carried by two electron charges as predicted by the BCS theory
for supercondtivity.

It is clear what happens when you cool down the conducting ring below
the critical temperature: Before the cooling, the material is
superconducting, and the magnetic flux can take any value. Cooling down
the ring, surface currents are induced such that an additional magnetic
field is created which together with the external field makes a flux,
which must be an integer multiple of \Phi_0.