Let $n-100= m^2$ and $n-63=k^2$, then we have $$(n-63)-(n-100)=k^2 - m^2 \implies 37 = (k+m)(k-m)$$

Because $37$ is a prime number which can only be factorized as $1\times 37$, Hence we must have $$k-m = 1, k+m=37\implies k=19, m=18$$

It follows that $n=m^2 + 100 = \boxed{424}$.