Take $(x, y)=(184,345)$. Then $x+y=23^{2}, x^{2}+y^{2}=391^{2}$, and $x^{3}+y^{3}=6877^{2}$. Remark. We need $x+y, x^{2}+y^{2}, x^{2}-x y+y^{2}$ to be perfect squares. We will find $a, b$ such that $a^{2}+b^{2}, a^{2}-a b+b^{2}$ are perfect squares, and then let $x=a(a+b)$ and $y=b(a+b)$. Experimenting with small Pythagorean triples gives $a=8, b=15$ as a solution. Remark. The smallest solution we know of not of the form \left(184 k^{2}, 345 k^{2}\right)$is$$(147916017521041,184783370001360)$$