Maybe because you apply it to something that is not a polynomial. Here is a way for Mathematica to automatically find the polynomial inside the expression:

findAndSquareComplete[expr_, var_] := expr /.
   (poly_ /; Exponent[poly, var] == 2) :> 
    squareCompletion[poly, var];
findAndSquareComplete[2 Sqrt[4 - 16/
     ((-4 + m)^2 + 4 m^2)], m]

Hmmm, it seems that Exponent does not check for polynomiality, unlike CoefficientList. We need a test for that:

squareCompletion[pol_, var_] /;
   PolynomialQ[pol, var] && Exponent[pol, var] == 2 := 
  Module[{var0, x}, var0 = SolveValues[D[pol, var] == 0, var][[1]];
   Collect[pol /. var -> var0 + x, x] /. x -> var - var0];
findAndSquareComplete[expr_, var_] := expr /.
   (poly_ /; PolynomialQ[poly, var] && Exponent[poly, var] == 2) :> 
    squareCompletion[poly, var];
findAndSquareComplete[(4 (12 - 8 m + 5 m^2))/(16 - 8 m + 5 m^2), m]

Why does the polynomial remain unchanged after using this function?