解:由题意,得$\begin{cases} x+y-2026≥0 \\ 2026-x-y≥0 \end{cases},$$\therefore x+y=2026。$
$\therefore \sqrt{3x+y-z-8}+\sqrt{x+y-z}=0。$又$\because \sqrt{3x+y-z-8}≥0,$$\sqrt{x+y-z}≥0,$
$\therefore \begin{cases} 3x+y-z-8=0 \\ x+y-z=0 \\ x+y=2026 \end{cases},$解得$\begin{cases} x=4 \\ y=2022 \\ z=2026 \end{cases}。$
$\therefore (z-y)^2=(2026-2022)^2=16$