The answer is "no" for $n=2$.

Choose $\Gamma=f(\partial M)$ to be a slight variation of flat nonconvex quadrangle.
You want to present $\Gamma$ as an intersection of two ruled surfaces which have no parallel tangent planes.

In other words, the set of normal unit vectors formed by two curves (say $\alpha$ and $\beta$) in $S^2$ do not intersect.
One of the curves ($\alpha$) is nearly constant, it corresponds to the half of $\partial[\mathop{Conv}\Gamma]$.
For the other, choose an involution $\phi$ on $\Gamma$ and connect by line $x$ to $\phi(x)$.
The choice can be made so that the obtained ruled surface is smooth.
Still we have a lot of freedom to ensure that $\beta$ avoids to go near $\alpha$.