Algebra

Do some manipulations and write the given equation as:-

$\underbrace{(x-y)^2+(y-z)^2+(z-x)^2}_{\Large \color{#D61F06}{\geq 0}}=\underbrace{-(x^2+y^2+z^2)}_{\Large\color{#D61F06}{\leq 0}}$

For $\large x,y,z\in\mathbb R$ , thus is only possible when both side are identically zero ( Since LHS is always non negative while RHS is always non positive ), so we conclude $x=y=z=0$ .

Hence, $\Large x+y+z=\huge\boxed 0$