Infinite square well separation of variables issue

When solving the time independent Schrodinger equation for the infinite square well in 2 or 3 dimensions (I’ll use 2 dimensions for brevity), we use separation of variables, first assuming that our solution is of the form $psi(x,y)=f(x)g(y)$ and we get to the point where we have that
$$frac{1}{f}frac{partial^2f}{partial x^2}+frac{1}{g}frac{partial^2g}{partial y^2}=frac{-2m}{hbar}E$$
My problems begin at this point. Clearly the two terms on the LHS of the above equation must be equal to a constant. Thus we have that
$$frac{1}{f}frac{partial^2f}{partial x^2}=C_x ,,,,,,,frac{1}{g}frac{partial^2g}{partial y^2}=C_y$$
But this is never how the problem is actually solved. Instead, it is always assumed that we actually have
$$frac{1}{f}frac{partial^2f}{partial x^2}=-k^2_x ,,,,,,,frac{1}{g}frac{partial^2g}{partial y^2}=-k^2_y$$
My problem is that this does not allow either of the terms of the form $frac{1}{f}frac{partial^2f}{partial x^2}$ to equal a positive constant. Yet one of them equalling a positive constant should still be a valid possibility. I am aware that in this example, $E>0$ because $E$ must always be greater than the value of the potential within the well. But even if we make the assumption that $E>0$, we can still have either one of $C_x$ or $C_y$ being positive because the sum of a positive $C_x$ and negative $C_y$ can still overall be negative provided $C_x<-C_y$. Thus the constants in my second equation should be able to be both positive or negative and still satisfy the Schrodinger equation. It seems to me that by assuming that both the terms $frac{1}{f}frac{partial^2f}{partial x^2}$ and $frac{1}{f}frac{partial^2g}{partial y^2}$ be equal to negative constants, we should lose out on many valid solutions to the schrodinger equation (all the cases where $ C_x<-C_y $ for example).

Any help on this issue would be most appreciated!