I am working on the same question as a poster here and I was able to prove the inequality by re-factoring to:
$$ (x-z)^2 + 4y(2-x-z) + 4y^2 \geq 0$$
and arguing that given the conditions, this holds (first and third terms are positive, for the second $y>0$ by definition and so is $2-x-z$).
However, I am also asked to determine when the equality holds. Given that I have a sum, I don't see a way to do that:
$$ (x-z)^2 + 4y(2-x-z) + 4y^2 = 0$$
Can someone explain the general direction I should move in to be able to prove the equality? The only bright thought was dividing by $4y$ and looking for a substitution, but this did not yield anything.