I need to compute the work done by the force $F(x,y,z)=(y-z)i+(z-x)j+(x-y)k$ on C such that C is the intersection between $x^2+y^2=4,~x+2z=2$.
$rot(F)=(-2,-2,-2)$, then I applied Green's Theorem and worked with polar coordinates: $0\leq r \leq 2$ and $0 \leq \theta \leq 2\pi$. The solutions yields $-12\pi$.
Can I conclude that as the rotor is negative, it indicates that the orientation of the curve is clockwise which implies work is $12\pi?$
The sign of the rotor does not determine the orientation of the curve. The orientation of the curve should be clarified by the given question. You should note that using Stoke's theorem (note that Green's theorem is the 2D version only) to convert the line integral to a surface integral, you need to orient the surface according to the orientation of the curve (using the right-hand rule). A correct orientation should give the correct sign.