A parametrization of solutions is
$x=2ad$
$y=2bd$
$z=2cd$
$t=a^2+b^2+c^2-d^2$
$w=a^2+b^2+c^2+d^2\ .$
It is easy to see that this generates all rational solutions if $a, b, c, d$ are rational numbers, and (consequently) all integers solutions up to a similarity factor, if $a, b, c, d$ are integers.
[edit] for what it's worth, here's something more symmetric (with the same rmk as before)
$x=-a^2+b^2+c^2+d^2-2a(b+c+d)$
$y=\phantom{-}a^2-b^2+c^2+d^2-2b(a+c+d)$
$z=\phantom{-}a^2+b^2-c^2+d^2-2c(a+b+d)$
$t=\phantom{-}a^2+b^2+c^2-d^2-2d(a+b+c)$
$w=\phantom{-}2(a^2+b^2+c^2+d^2)\ .$
