smc

We see that the vertex of the quadratic function $y=x^2+a^2$ is $(0,a^2)$. The y-intercept of the line $y=x+a$ is $(0,a)$. We want to find the values (if any) such that $a=a^2$. Solving for $a$, the only values that satisfy this are $0$ and $1$, so the answer is $\boxed{2}$