There are no points other than the trivial points $(n, \pm n, 0)$, $(n,0,\pm n)$ and $(0,n,\pm n)$, and in particular, no integer points with all coordinates positive. For example, taking the affine part $f(\alpha,\beta, 1)$ of the above curve $f(a,b,c)$ and then setting $x:=\alpha+\beta$, $y:=\alpha\beta$, one verifies that the curve defined by $x$ and $y$ (of which the original (genus 4) curve is a cover) is of genus $1$, i.e., elliptic, and has (finite) Mordell-Weil group $\mathbb{Z}/6\mathbb{Z}$ (I asked Magma for this, returning this curve: http://www.lmfdb.org/EllipticCurve/Q/14/a/5 - although there may well be a more direct argument). These are all either points at infinity or having $y(=\alpha\beta)=0$, i.e., already this subcover of the original curve has no ``good" points.