Of all the North American CX gateways, you will get the best lounge at JFK as a first class passenger.
I wouldn't bother with the AA's transcontinental flagship service. CX's transcontinental service (JFK-YVR) will be markedly superior than AA's.
If you took the AA transcontinental flagship service, you will be disappointed with the AAdmirals Club at JFK and the oneworld lounges at Tom Bradley or the Sakura Lounge at SFO.
As Carfield suggested, I recommend taking CX to YVR, check-in at the Fairmont Vancouver Airport (Nice hotel. Don't forget to get a spa treatment at the hotel. Spa treatments in Canada are cheaper. My first choice would be the Four Seasons, but you would only have a 12 hour layover and it wouldn't be worth the hassle to go downtown).