Elementary manipulations designed to solve q(x,y,z)=0 for x lead to an equation quartic in x. Hence, one expects four solutions for x=x(y,z).
In Maple 13, "solve" does not return any explicit solutions for x. In Maple 14 it does, but at least 3 of the 4 branches don't seem to satisfy the original equation. Hence, at least one of the local minima supposedly found by analyzing these branches doesn't satisfy the original equation.
It appears that the minimum value of -4 is correct, but the analysis based on an explicit solution for x=x(y,z) is highly suspect.
The function q was given as
q:=(x,y,z)-> -5 + ((x*y*z/z+1-x)+ sqrt( (x*(x+2*z^2-2*y)/x+3*z-1))/ (4*x*z^2-y));
Taking this literally means that "z" in the product x*y*z cancels out, and that under the radical, the factor of "x" also cancels out. (I have to wonder if the author meant to enclose some of these divisors with parentheses.) Hence, the radicand is actually x+2*z^2-2*y+3*z-1. For real values, this expression has to be nonnegative. The feasible region lies above a surface x(y,z) easily found explicitly. Over the square 0<=y,z<=1, the lowest point on this surface is (x,y,z)=(-4,0,1).