= Solution
\b[False.] Take $X=\mathbb R$ and $A=B=\{0\}$. Both complements are disconnected: their negative and positive halves form a separation. But
$$
(X\times X)\setminus(A\times B)=\mathbb R^2\setminus\{(0,0)\}
$$
is <path-connected>, hence <connected>. To see <path connectedness> directly, move any nonzero point radially to its normalization on the unit circle; join the two normalized points by a circular arc; then reverse the radial path for the second point. Every segment of this path avoids the origin. Thus disconnected coordinate complements do not force a disconnected complement of their product.
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