Choice A is the correct answer.

1. Substitute $y=x^2$ into $x=y^2$: $x = (x^2)^2 = x^4$.
2. $x^4 - x = 0$.
3. Factor: $x(x^3 - 1) = 0$.
4. Real solutions: $x = 0$ and $x = 1$.
5. Points: $(0,0)$ and $(1,1)$.

Choice B is incorrect. Choice C is incorrect. Choice D is incorrect.