Higgs mechanism and neutral fields Consider a Lagrangian $L(\phi,A_{\mu})$ with $\phi$ being some scalar field and $A_{\mu}$ some dynamical U(1) gauge field that minimally couples to $\phi$. 
Under a global U(1) symmetry the field $\phi$ transforms as 
$$
\delta\phi=i\epsilon q \phi.
$$
The field $\phi$ is said to be charged (with charge q) under the gauge field $A$. 
In a Higgs phase we have that $|\phi(x,t)|\neq 0$. In particular we can fix a gauge so that  $|\phi(x,t)|=\Phi(x,t)$ is real. Then we consider small fluctuations 
$\Phi(x,t)=\Phi_{0}+\delta\Phi$ and integrate them out to obtain an effective theory in which the gauge field A is massive. 
My question: It seems to me as though the requirement that $\phi$ is charged enters 
when integrating out the small flunctuations, because if $\phi$ were neutral (i.e. q=0) there wouldn't be any flucutations that one can integrate out and hence one wouldn't obtain a massive term for the gauge field in the Lagrangian. Is this correct? If not where does the requirement for $\phi$ to be charged enter in the argument? And: Does the requirement for the matter field to be charged with respect to the corresponding gauge field carry over without difficulties to the non-abelian case? 
I am looking forward to your responses! 
 A: Like you said "$A_\mu$ some dynamical $U(1)$ gauge field that minimally couples to $\phi$". It means that the covariant derivative is :
 $$D_\mu \phi = \partial_\mu \phi + iqA_\mu\phi$$ 
with $q$ the $U(1)$ charge of the scalar field. As a consequence, if $\phi$ is not $U(1)$ charged you will not have the second term in the covariant derivative and hence $D_\mu$ is equivalent to the standard derivative $\partial_\mu$. When the scalar field $\phi$ is VEVed with fluctuations around this VEV : 
$$\phi = \frac{(v+h_1)+ih_2}{\sqrt{2}}$$
and that you compute the kinetic term $(D_\mu \phi)^\dagger D_\mu \phi$, you will get a mass term for $A_\mu$ which is :
$$\frac{1}{2}q^2v^2A_\mu A^\mu$$
which is proportional to $q$ the $U(1)$ charge of $\phi$ and this term appears in the contraction of the second terms of $(D_\mu \phi)^\dagger$ and $D_\mu \phi$. If these second terms was not here, i.e. if the scalar field $\phi$ was not $U(1)$ charged, then the gauge field $A_\mu$ can't get a mass.
Finally, the requirement that $\phi$ is $U(1)$ charged enters when you require a non-trivial minimal coupling between $A_\mu$ and $\phi$.
$\textbf{EDIT}$ : 
If you've already seen the Glashow-Weinberg-Salam (GWS) Model, the analogy is that the Higgs field is $SU(2)_L\times U(1)_Y$ charged (because it's an $SU(2)$ doublet and it has an hypercharge $Y$). Thus, when the Higgs field is VEVed, the gauge fields acquire a mass. 
A: Let me answer your questions, albeit slightly indirectly. We start with a local $U(1)$ symmetry, i.e. a gauge symmetry, for a Lagrangian describing a scalar, $\phi$, and a gauge boson $A_\mu$. You write global. A global symmetry requries no gauge bosons, because its continuous parameter $\epsilon\neq\epsilon(x)$ commutes with derivatives $\partial_\mu$. The gauge symmetry prohibits a mass for the vector boson. Supposing that the scalar obtains a non-zero vacuum expectation value (VEV)...


*

*If the scalar is charged under the $U(1)$, the $U(1)$ is said to be
spontanouesly broken or hidden, because it is no longer manifest in
our Lagrangian. In particular, we now have a massive gauge boson. We
do not need to integrate out the fluctuations about the VEV (i.e. the 
Higgs   boson) to see that the gauge
boson has obtained a mass.

*If the scalar is not charged under the $U(1)$, the symmetry is
unbroken, and the vector boson remains massless. These arguments do
transfer to the non-Abelian cases. Indeed, the VEVed part of the
Standard Model scalar doublet is electically neutral, so does not
break the $U(1)_{em}$ electromagnetism.
