Using a compiled function inside NIntegrate gives "CompiledFunction::cfsa" message

It's because FFc is being passed x and Ef symbolically, at first. To cure the problem, add an intermediate function between FFc and NIntegrate, such as

f[x_?NumericQ, Ef_?NumericQ] := FFc[x,Ef]

then

foo[Ef_?NumericQ] := NIntegrate[f[x, Ef] , {x, -EBoundary, EBoundary}]

This answer addresses Ajasja's question in rcollyer's answer:

Do you perhaps know why adding "RuntimeOptions" -> {"EvaluateSymbolically" -> False} to FFc does not cure the problem?

This is due to the fact that the methods inside NIntegrate[] (attempt to) perform a preliminary symbolic analysis, which can be helpful for integrands composed of built-in mathematical functions, but not terribly useful for compiled functions. Thus, one has to turn it off in this case, in addition to enabling the compilation options mentioned by Ajasja.

Here is a quick demonstration:

FFc = Compile[{{x, _Real}, {EF, _Real}}, 
              If[x > EF, 0., If[x == EF, 0.5, 1.]], 
              "RuntimeOptions" -> {"EvaluateSymbolically" -> False}];

With the default setting, we get this (Mathematica 10.4):

NIntegrate[FFc[x, 3.2], {x, -6.5, 6.5}]

CompiledFunction::cfsa: Argument -x at position 1 should be a machine-size real number.
CompiledFunction::cfsa: Argument -x at position 1 should be a machine-size real number.
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the
following: singularity, value of the integration is 0, highly oscillatory integrand, or
WorkingPrecision too small.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9
recursive bisections in x near {x} = {3.22441}. NIntegrate obtained 9.699759342263453`
and 0.0005405922634428764` for the integral and error estimates.

before 9.69976 is returned. In contrast, if we disable symbolic processing like so:

NIntegrate[FFc[x, 3.2], {x, -6.5, 6.5}, Method -> {Automatic, "SymbolicProcessing" -> 0}]

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the
following: singularity, value of the integration is 0, highly oscillatory integrand, or
WorkingPrecision too small.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9
recursive bisections in x near {x} = {3.22441}. NIntegrate obtained 9.699759342263453`
and 0.0005405922634428764` for the integral and error estimates.

we no longer get the CompiledFunction::cfsa message.