= Solution
Use the common <filtration> in which the two driving processes are independent <Brownian motions> and the solutions are <adapted>. Let $Z=X+Y$, and define <predictable> bounded weights
$$
h_1=\mathbf1_{\{Z>0\}}\sqrt{X/Z}+\mathbf1_{\{Z=0\}},\qquad
h_2=\mathbf1_{\{Z>0\}}\sqrt{Y/Z},
$$
with the ratios defined only on $\{Z>0\}$. The nonnegative solutions have $X=Y=0$ when $Z=0$. Set
$$
B_t=\int_0^t h_1(s)dB_s^{(1)}+\int_0^t h_2(s)dB_s^{(2)}.
$$
Independent Brownian drivers have zero cross-variation. Since $h_1^2+h_2^2=1$ even on the zero set, $[B]_t=t$, and the <Lévy characterization of Brownian motion> makes $B$ a <Brownian motion>. Moreover $\sqrt Z h_1=\sqrt X$ and $\sqrt Z h_2=\sqrt Y$ everywhere. Adding the two original equations therefore proves <additivity of independently driven squared Bessel processes>:
$$
\boxed{dZ_t=2\sqrt{Z_t}\,dB_t+(\alpha+\beta)dt,\qquad \gamma=\alpha+\beta.}
$$
The unit-vector fill on $\{Z=0\}$ is essential: simply dividing by $\sqrt Z$ there would leave the Brownian construction undefined.
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