Answer: 146

Small Hint:

Let $s=x+y$ and $p=xy$, and rewrite both given equations using $s$ and $p$

Big Hint:

The two equations determine $s+p$ and $sp$, so $s$ and $p$ are roots of one quadratic

Solution:

Let $s=x+y$ and $p=xy.$ The equations become $s+p=71$ and $sp=880,$ so $s$ and $p$ are the roots of $$\begin{aligned}t^2-71t+880&=0,\\ (t-16)(t-55)&=0.\end{aligned}$$ Because $x$ and $y$ are positive integers, $s=16$ and $p=55$; indeed, $x=5$ and $y=11.$ Therefore $$\begin{aligned}x^2+y^2&=s^2-2p\\&=16^2-2(55)=146.\end{aligned}$$