  We report neutron diffraction results for the body-centered-tetragonal series
Eu(Co$_{1-x}$Ni$_x$)$_{2-y}$As$_2$, $x=0.10$, $0.20$, $0.42$, and $0.82$,
$y\leq0.10$, that detail changes to the magnetic ordering with nominal hole
doping. We report the antiferromagnetic (AFM) propagation vectors, magnetic
transition temperatures, and the ordered magnetic moments. We find a
nonmonotonic change of the AFM propagation vector with $x$, with a minimum
occurring at the tetragonal to collapsed-tetragonal phase crossover. For
$x=0.10$ and $0.82$ we find $c$-axis helix ordering of the Eu magnetic moments
(spins) similar to $x=0$ and $1$, with the spins oriented within the
$ab$-plane. For $x=0.20$ and $0.42$ we find higher-temperature $c$-axis FM
order and lower-temperature $c$-axis cone order. Using the extinction
conditions for the space group, we discovered that the Eu spins are ordered in
the higher-temperature $c$-axis FM phase for intermediate values of $x$,
contrary to a previous report suggesting only Co/Ni spin ordering. Although we
cannot directly confirm that the Co/Ni spins are also ordered, we suggest that
$c$-axis itinerant-FM ordering of the Co/Ni spins could provide a molecular
field that drives FM ordering of the Eu spins, which in turn provides the
anisotropy for the lower-temperature $c$-axis cone order.
