Wolfram Alpha solves the two given equations with

y=(-s*sqrt(3)*sqrt(4-x^2)-x)/2

z=(s*sqrt(3)*sqrt(4-x^2)-x)/2

where s= +/- 1

The program (in its final form in terms of the rangeof x):

mx=0;

for x=1.97859905:.0000000001:1.97859906

  for s=[-1 1]

    y=(-s*sqrt(3)*sqrt(4-x^2)-x)/2;

    if isreal(y)

      z=(s*sqrt(3)*sqrt(4-x^2)-x)/2;

      if isreal(z) && isreal(y)

        v=abs((x-y)*(x-z)*(z-x));

        

        if v>mx

          disp([x y z v])

          mx=v;   

          mxx=x;

          mxy=y;

          mxz=z;

        end

      end

    end

  end

end

disp([mxx mxy mxz mx]);

finds the value is maximized at x =~ 1.9785990527 where the value is approximately 28.1627614887374.

x=1.9785990527 y=-0.736595467239353 z=-1.24200358546065

 |(x-y)(y-z)(z-x)|=28.1627614887374

This is very close to the value of x=2 where y and z become complex, and there may be some strange things happening, as a plot that allows complex values looks as if it's monotonically increasing with x. However, computation with extended precision, i.e. Matlab's vpa type, shows decreasing values between this value of x and x=2.