The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I couldn't find any reference for this claim.

So my question is: Are there relatively simple examples of amalgamation of ccc algebras that collapse the continuum? If there aren't any simple examples, then a reference for a harder example would be welcome as well.