It's hip to be a square!

With $y = M^{2}$ for some integer $M$ , we can rewrite the equation as

$M^{2} = (N + 4)^{2} - 19 \Longrightarrow (N + 4)^{2} - M^{2} = 19.$

Now the only perfect squares that differ by $19$ are $10^{2} = 100$ and $9^{2} = 81$ . Thus we can either have

$N + 4 = 10 \Longrightarrow N = 6$ , or

$N + 4 = -10 \Longrightarrow N = -14.$

Thus the sum of possible values of $N^{2}$ is $6^{2} + (-14)^{2} = \boxed{232}$ .

We can write it as : (N+4)²=M² +19
and the only solution is (N+4)²=100
SO, N can either be 6 or - 14
And N² can either be 36 or 196 Therefore, the answer is 36+196=232