I guess you also have to assume something about the subset of $R^d$ of allowed values for $z$.
If it is just a singleton, then writing $e_n=(x_n-z)/\|x_n-z\|_2=(e_{n,j})_{j=1}^d$, we have $$E_\sigma \left\|\sum\sigma_ne_n\right\|_2=E_\sigma\sqrt{\sum_j \left(\sum_n\sigma_ne_{n,j}\right)^2}\le\sqrt{E_\sigma\sum_j\left(\sum_n\sigma_ne_{n,j}\right)^2}$$ $$%=\sqrt{\sum_jE_\sigma\left[\left(\sum_n\sigma_ne_{n,j}\right)^2\right]} =\sqrt{\sum_j\mathrm{Var}_\sigma\left[\sum_n\sigma_ne_{n,j}\right]} =\sqrt{\sum_j\sum_n\mathrm{Var}_\sigma\left[\sigma_ne_{n,j}\right]}$$ $$=\sqrt{\sum_j\sum_ne_{n,j}^2}=\sqrt{\sum_n \|e_n\|_2^2}=\sqrt{N}.$$