To simplify the statement, I will use the following notation for real vectors and matrices. Let $\epsilon_i = \boldsymbol{e}_i^T \boldsymbol{\epsilon}$, where $\boldsymbol{e}_i \in \mathbb{R}^N$. Sometimes $\boldsymbol{e}_i$ is called the $i$-th standard basis vector; it has all elements equal to 0 except the $i$-th element equal to 1. We have
$$ E(\epsilon_i^2) = E[(\boldsymbol{e}_i^T (I - G) \boldsymbol{x})^2] = \boldsymbol{e}_i^T (I - G)(I - G^T) \boldsymbol{e}_i$$
or
$$
E(\epsilon_i^2) = \|(I - G^T) \boldsymbol{e}_i\|_2^2.
$$ 
By the way, the given condition $E\{\boldsymbol{x}^H\boldsymbol{x}\} = I$ should be $E\{\boldsymbol{x}\boldsymbol{x}^H\} = I$.