The answer is no. E.g., let $U=(0,1/2)$ and 
$$\zeta=2\times 1_{(0,1/2)^2}+2\times 1_{(1/2,1)^2};$$
that is, the joint distribution of $X,Y)$ is the half-and-half mixture of the uniform distributions on the the squares $(0,1/2)^2$ and $(1/2,1)^2$. Then 
$$P_{X,Y}(U\times U)=\frac12\ne\frac12\times\frac12=P_X(U)P_Y(U).$$