I've not had the privilege of flying F yet, so I can't really comment on Flagship First dining.
Breakfast offerings at JFK Flagship buffet are typically western. The times I've been there in the morning they've offered things like eggs benedict, american bacon, sausage and french toast.
JFK Flagship is great but it does tend to get a little crowded. To the right of the main lounge, when entering, is an area called Flagship Bridge. It offers casual sit down service with a menu of small plates and a bar. Most pax tend to go the the main lounge so the Bridge area is usually very quiet with plenty of seating. The operating hours are more limited, I think it's open 10AM-10PM. If I'm taking CX865, I usually sit there first and have a small dinner then head on over to the main lounge area for some tea and dessert and grab a bottle of water before heading to the gate.
JFK Flagship is a great lounge but I wouldn't put it above any of the CX HK lounges, except maybe the Deck. As stated above, CX J lounges offer a lot of Asian comfort food which can be carb heavy as well as the western choices like pizza and sandwiches. The food choices at Flagship are more limited and tend towards a western meal where you can choose a hunk of meat with starch and vegetable sides and it's all prepared quite well.
Last time I flew CX865 it was departing from gate 47. I headed over there and found there was a delay of 30 minutes or so. Just for kicks I headed over to the AA Admirals lounge not far away. It was gigantic and empty. Not much food available as many of the stations were closed but I wasn't looking for that. As crowded as Flagship was this was the opposite. Staff told me this Admirals lounge doesn't get very crowded. It won't knock your socks off but if you're looking for some privacy it can be had there.