Free product of Boolean algebras

Translating this to Boolean spaces, you are looking for a Boolean space X which is not second countable, but cannot be written as a product of two factors of the same type (i.e., not second countable).

Have you considered the compact space $[0,\omega_1]$? It is certainly not the product of two uncountable spaces, as such a product would contain two almost disjoint closed uncountable sets. On the other hand, a countable Boolean space cannot have uncountably many clopen sets.

By Theorem 15.14 of the Boolean algebra handbook, the interval algebra of the real numbers is such an algebra B.

Reference: S. Koppelberg. Handbook of Boolean algebras. Vol. 1. Edited by J. D. Monk and R. Bonnet. North-Holland Publishing Co., Amsterdam, 1989.