jmc

We can write the given expression as $$\begin{array}{rl}& (x^2-4x+4)+(x^2-2xy+y^2)+(y^2-4yz+4z^2)+(z^2-2z+1)+10\\ & =(x-2{)}^2+(x-y{)}^2+(y-2z{)}^2+(z-1{)}^2+10\end{array}$$The minimum value is then $\boxed{10},$ which occurs when $x=2,$ $y=2,$ and $z=1.$