If the autocorrelation function of a zero-mean (stationary) Gaussian white noise
process is $R_X(\tau) = E[X(t)X(t+\tau)] = K\delta(\tau)$,
then the result $Y(t)$ of passing $X(t)$ through a filter with impulse response
$h(t)$ is also a zero-mean stationary Gaussian process with
autocorrelation function (and autocovariance function) given by
$$R_Y(\tau) = E[Y(\lambda)Y(\lambda+\tau)]
= K\int_{-\infty}^{\infty} h(t)h(t+\tau)\, \mathrm dt.$$
The above applies to Gaussian white noise as engineers understand
the term. But, as I noted in this unanswered question, mathematicians
also use the term (stationary) white noise to mean a process
whose autocovariance function is given by
$$\text{cov}(X(t), X(t+\tau))
= \begin{cases} \sigma^2, &\tau = 0,\\
0, &\tau \neq 0, \end{cases}$$
in which case the description above does not apply.
I would suppose that nonstationary white noise would mean
$\text{var}(X(t))= \text{cov}(X(t), X(t))$ varies with $t$
instead of having fixed value $\sigma^2$ but I am not too
sure about this.