Chris Heunen's comment under the OP can be turned into a proof. Suppose the category of compact Hausdorff extremally disconnected spaces has binary products. Let $X \times Y$ denote the product in that category. If $|X|$ denotes the underlying set, then of course the canonical map

$$|X \times Y| \to |X| \times |Y|$$

is an isomorphism, because $|X| \cong \hom(\ast, X)$ where $\ast$ is the one-point space, i.e., the underlying set functor is representable and representables preserve products.

Chris observes that the ordinary product space $X \times_{Top} Y$ of two compact Hausdorff extremally disconnected spaces need not be extremally disconnected. However, under our supposition we would have a continuous comparison map

$$X \times Y \to X \times_{Top} Y$$

in $Top$ which is a bijection at the level of the underlying sets. Being a continuous bijection between compact Hausdorff spaces, it is a homeomorphism, and this contradicts Chris's observation.

equivalencepreserves products. – Andrej Bauer Jan 11 '11 at 17:59and such an object does not existassuming AC (dx.doi.org/10.1007/BF02757883) – Adam Jan 11 '11 at 18:44