*Answer.* The only possible values are $8$ and $-1$.

We add $1$ to the equations and obtain
$$\frac{ x + y + z }{ z } = \frac{ x + y + z }{ x } = \frac{ x + y + z }{ y }.$$
For $x + y + z = 0$, this is clearly true, and the expression in the problem statement becomes
$$\frac{ (-z)(-x)(-y) }{ xyz } = -1.$$
For $x + y + z \neq 0$, we get
$$\frac{ 1 }{ z } = \frac{ 1 }{ x } = \frac{ 1 }{ y }.$$
and therefore $x = y = z$.
In this case, our expression becomes $8$.
The values $-1$ and $8$ are attained because any triple with $x + y + z = 0$ resp. $x = y = z$ that does not contain a zero works.
(Theresia Eisenkölbl) ☐