Strongest chloride complex as predicted by HSAB theory
The link to your handout explains why $\ce{Fe^{3+}}$ (and it is $\ce{Fe^{3+}}$ in $\ce{FeCl4^{-}}$) is a harder acid than $\ce{Co^{3+}}$ or $\ce{Cu^{2+}}$ or $\ce{Ni^{2+}}$.


  Hard acids (in context, HA) are characterized by (s,f blocks, left side of d block in higher OS's)


Iron is the furthest to the left of the four metals you mention. Thus, it has fewer electrons in its $d$ shell to balance the positive charge:


$\ce{Fe^{3+}}$ is $\ce{[Ar]}3d^5$
$\ce{Co^{3+}}$ is $\ce{[Ar]}3d^6$
$\ce{Ni^{2+}}$ is $\ce{[Ar]}3d^8$
$\ce{Cu^{2+}}$ is $\ce{[Ar]}3d^9$


and 


  Low electronegativity (χ) of the acidic atom. A value in the range 0.7-1.6 is typical of hard acids;


Iron has the lowest electronegativity of the four metals:


$\ce{Fe}$ 1.83
$\ce{Co}$ 1.88 
$\ce{Ni}$ 1.91 
$\ce{Cu}$ 1.90


Iron is the hardest Lewis acid of the metal cations you list.

As for $\ce{Ni^{2+}}$, it does form a complex $\ce{NiCl4^{2-}}$ with $\ce{Cl-}$. Here is a Google search for it.