The underlying problem here is proving that the passenger flew the outbound. If she did, then AA improperly cancelled her ticket and she was denied boarding and due EC 261/2004 compensation of EUR 600. If she did not fly the outbound (or cannot prove it), then AA properly cancelled the remaining segment of her ticket, she was properly denied boarding and due nothing.
Proving that one took a flight can be hard. BP's themselves are meaningless, whether on paper or print. But, nonetheless, because carriers foolishly accept them as proof, it makes sense to hang onto them until all issues relating to the flight have been settled. e-BP's are fine. AA's remain in one's wallet until deleted and simply have a note on the code to the effect that the pass has expired. Alternatively, one may always save a PDF of the BP.
If one lacks a BP, it is all a matter of gathering other items which might tend to prove this. Bear in mind that the passenger count for the aircraft had to match before it closed and pushed back. Thus, somehow AA had to account for a person being on its aircraft.