My experience, in 2011. We arrived in F from JFK, spent 22 hours in HKG and were flying out in J to DPS (Bali), this was a reward ticket using American miles, I did not have any kind of status with CX or AA. We were allowed to use F lounge the morning of our HKG-DPS flight as it was part of the same reservation for F-Class and this airplane only had J-Class available. I don't know if that changed, but at least that is my experience from 3 years ago.
I am doing a similar flight later this year JFK-HKG (22 hour stop) - SIN, and it is in F-class and J-Class, it is a reward ticket using AA miles, so I assumed we will be allowed to use F-Class lounge the morning of HKG-SIN flight, I hope that policy did not change.
I hope this helps.
Cheers,
PedroNY