Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)
You have to look at the autocorrelation function of $y(t)$:
$$R_y(\tau)=E\{y(t)y(t+\tau)\}\tag{1}$$
with
$$y(t)=(h_A\star a)(t) - (h_B\star b)(t)\tag{2}$$
where $\star$ denotes convolution. If you write out $(2)$ with integrals and plug it into $(1)$ then, with the given assumptions on $a(t)$ and $b(t)$, you'll see that the mixed terms with the negative sign are zero, and the two remaining terms with a positive sign become
$$R_y(\tau)=\sigma_a^2r_{h_A}(\tau)+\sigma_b^2r_{h_B}(\tau)\tag{3}$$
where $\sigma_a^2$ and $\sigma_b^2$ are the variances of the processes $a(t)$ and $b(t)$, respectively, and $r_{h_A}(\tau)$ and $r_{h_B}(\tau)$ are the deterministic autocorrelation functions of $h_A(t)$ and $h_B(t)$, respectively. Finally, taking the Fourier transform of $(3)$ gives
$$S_y(\omega)=\sigma_a^2|H_A(j\omega)|^2+\sigma_b^2|H_B(j\omega)|^2\tag{4}$$
for the power spectrum of $y(t)$.