Low pass equivalent of bandpass white noise
Note that for the complex noise envelope $z(t)=x(t)+jy(t)$, the autocorrelation $R_z(\tau)$ is defined by (cf. Eq. $(4.1.47)$ in Proakis)
$$R_z(\tau)=\frac12E\big\{z^*(t)z(t+\tau))\big\}=\frac12\big[R_x(\tau)+R_y(\tau)\big]+j\frac12\big[R_{xy}(\tau)-R_{yx}(\tau)\big]\tag{1}$$
As shown in the chapter you refer to, for the real-valued bandpass noise $n(t)$ to be stationary, the following must be true:
$$\begin{align}R_x(\tau)&amp;=R_y(\tau)\tag{2}\\R_{xy}(\tau)&amp;=-R_{yx}(\tau)\tag{3}\end{align}$$
Plugging $(2)$ and $(3)$ into $(1)$ we obtain
$$R_z(\tau)=R_x(\tau)+jR_{xy}(\tau)\tag{4}$$
Note that this expression is different from the one you suggested in your question.
If the power spectrum of $z(t)$ is even, as is the case for bandpass white noise, the autocorrelation function $R_z(\tau)$ must be real-valued, and, consequently, $R_{xy}(\tau)=0$. But note that we don't require $R_y(\tau)=0$.
As a final note, defining a complex envelope is only useful for bandpass processes, such as bandpass white noise, but not for ordinary white noise, because the latter is no bandpass process.