Vector Calculus · Superchallenge problem

The plane $x + y + z = 1$ cuts the sphere $x^2 + y^2 + z^2 = 1$ in a circle $C$, oriented counterclockwise as seen from the point $(1,1,1)$. Evaluate $\displaystyle\oint_C y^2\,dx + z^2\,dy + x^2\,dz$. Give your answer in simplest radical form in terms of $\pi$.