Expanding given equation
$x^2 + y^2 + z^2 - \frac{2}{3}(x+y+z) + \frac{1}{3}$
$\quad = x^2 + y^2 + z^2 - \frac{2}{3} + \frac{1}{3}$
$\quad = (x+y+z)^2 - \frac{1}{3} - 2(xy + yz +xz)$
$\quad = 1 - \frac{1}{3} - 2(xy + yz +xz)$
$\quad = \frac{2}{3} - 2(xy + yz +xz)$
Now to maximize it, we need to minimize $(xy + yz + xz)$. As all $x,y$ and $z$ are non-negative $ xy + yz + xz \geq 0.$ So, the maximum value is $\frac{2}{3}$.
Correct Option: B.