Answer
False.
Work Step by Step
$f(x+h)$ is created when you input $x+h$ into $f(x)$. This leaves $$f(x+h)=(x+h)^2=(x+h)(x+h)$$ Use the distributive property to get $$f(x+h)=x(x+h)+h(x+h)=x^2+hx+hx+h^2=x^2+2hx+h^2$$ This result isn't equal to $x^2+h^2$, so the statement is false.