Squares, squares everywhere!

$x² ± y² = n$

$y²\equiv 0, 1, 4\pmod{8}$ → $x²\equiv 0, 1, 4\pmod{8}$

In module 8

$0 + 1 = 1$ , $1 + 1 = 2$ , $4 - 1 = 3$ , $0 + 4 = 4$ , $1 + 4 = 5$ , $0 - 1 = 7$ , $4 + 4 = 8$

$x² ± y² ≠ 8k + 6$

$8(-1) + 6 = -2$ , $8(0) + 6 = 6$ , $8(1) + 6 = 14$ , $8(2) + 6 = 22$ , $8(3) + 6 = 30$

Finally $6 + 14 + 22 + 30 = 72$

The first four such non negative integers are $6$ , $14$ , $22$ , and $30$ , and their sum is $\boxed{72}$