Follow user9325's suggestion about completing the squares, and then (look up and) apply the theory of Pell equations.
Edit: OK, I guess you didn't get anything out of user9325's suggestion, so I'll take it up for you.
$N=(x^2+y)/(x+y^2)$, $Ny^2-y=x^2-Nx$, $U^2-NV^2=1-N^3$ where $U=2Ny-1$, $V=2x-N$. This has the solution $U=-1$, $V=\pm N$. The solution $U=-1$, $V=N$ corresponds to $x=N$, $y=0$, which already shows that there's a solution for each $N$, but maybe that's too trivial. Then take any solution to $a^2-Nb^2=1$ and you get another solution, $U=-a\pm bN^2$, $V=-b\pm aN$. Now $a^2-Nb^2=1$ has lots of solutions - that's the Pellian I alluded to. For $y$ to be an integer, you need $a\equiv 1\pmod N$, so you have to study enough of the theory to see if that can be made to happen.
If $N$ is a square, say, $N=m^2$, then the Pellian doesn't apply, but you have something simpler; $(U+mV)(U-mV)=1-m^6$. Now you'll get at most finitely many solutions, since there are only finitely many ways to factor $1-m^6$. Here's one example; take $N=4=2^2$ so $m=2$ and $1-m^6=-63$; take $U+2V=63$, $U-2V=-1$ to get $U=31$, $V=16$; then $x=10$, $y=4$.