Determining Kohn-Sham and Hartree Fock virtual orbitals: The underlying field
That is true: in the Kohn–Sham model electrons both in occupied and in virtual orbitals are "moving" in the field $n-1$ electrons, while in the Hartee-Fock model electrons in occupied orbitals are "moving" in the field $n-1$ electrons, but electrons in virtual orbitals are "moving" in the field all $n$ electrons. And this is simply "by construction": you just have to look at the very definitions of these models. So, in short: 



In the Kohn–Sham model, it is the exchange-correlation hole that by definition "excludes" a single electron from equations describing each and every orbital. 

If you look at the Kohn–Sham equations
$$
\left(-{\frac  {\hbar ^{2}}{2m}}\nabla ^{2}+v_{{{\rm {eff}}}}({\mathbf  r})\right)\phi _{{i}}({\mathbf  r})=\varepsilon _{{i}}\phi _{{i}}({\mathbf  r}) \, ,
$$
where
$$
v_{\rm {eff}}(\mathbf {r} )=v_{\rm {ext}}(\mathbf {r} )+e^{2}\int {\rho (\mathbf {r} ') \over |\mathbf {r} -\mathbf {r} '|}d\mathbf {r} '+v_{{{\rm {xc}}}} \, ,
$$
it is last term of the Kohn–Sham effective potential, the exchange-correlation potential,
$$
v_{{{\rm {xc}}}}({\mathbf  r})\equiv {\delta E_{{{\rm {xc}}}}[\rho ] \over \delta \rho ({\mathbf  r})} \, ,
$$
that causes the above mentioned feature of Kohn–Sham orbitals.



For the case of the Hartree-Fock equations
$$
  \hat{F} \phi_i({\mathbf r}) = \epsilon_i \phi_i({\mathbf  r}) \, ,
$$
the summation in the Fock operator by the very definition of the model runs over all occupied orbitals only,
$$
  \hat{F}
  =
  \hat{H}^{\text{core}}
  +
  \sum_{j=1}^{n}[\hat{J}_{j} - \hat{K}_{j}] \, .
$$
For occupied orbitals, in the Hartree–Fock equation defining some particular (occupied) $i$-th orbital the Coulomb and exchange contribution from this orbital itself perfectly cancel each other, i.e. $\hat{J}_{j} \phi_i = \hat{K}_{j} \phi_i$ when $j=i$. Thus, in the Hartree-Fock model an electron from any occupied orbital is also moving in the field $n-1$ electrons. And this is of course physically more than reasonable: at the end of the day electron should not interact with itself.

But for virtual orbitals the situation in the Hartree-Fock model is different from that in the Kohn-Sham one: for any virtual orbital $\phi_k$ the sum in the Hartree-Fock equation defining it still runs over occupied orbitals only, but none of the terms $\hat{J}_{i} \phi_k$ and $\hat{K}_{i} \phi_k$ this time cancel each other since $k$ is simply grater than $n$.