<s>[INST] Is there any solution such that.. Is there any positive integer solutions of $x$, $y$, $z$ such that both $$\frac{x^2+y^2+z^2}{x+y+z}$$ and $$\frac{x^2+y^2+z^2}{xyz}$$ are integers?
Edit: Forgot to mention that $x$, $y$, $z$ are distinct positive integers.
 [/INST] Special case
Well, if $x = y = z$ then your equations become
$$\frac{3x^2}{3x} = x$$
$$\frac{3x^2}{x^3} = \frac{3}{x}$$
If they are coupled, then you notice that $x = 1$ is the simplest result. There is also $x = 3$: $3/x$ is an integer if and only if $x = 1$ or $x = 3$.
Unless you count negative integers too, in that case you have also $x = y = z = -1$ and $x = y = z = -3$
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