Potential energy with constraints moving body

I know that for conservative forces $\vec{F}=-\nabla{U}.$ Let's consider the case of gravitational potential energy, I know that $U=mgy$ . Just to check: $\vec{F}=-\nabla{U}=(0,-mg)$ perfect! Now, let's suppose that the body is only allowed to move on the line $y=x$ . The potential energy is as before $U=mgy$ , but now I could also write it as $U=mgx$ . Then I want the force, let's use the second equivalent expression: $\vec{F}=-\nabla{U}=(-mg,0)$ definitely not! What happened? I conclude that I must be careful when deriving potential energy. How? Other similar problems: I force the body to move along a specific constraint and I manage to express the potential as function of say $z$ . Am I allowed to conclude that $F_x=0$ because derivative of a function that not depends on $x$ is zero? What is the point behind these problems?