I would imagine that the problem would disappear as quickly as it came up if a few Captains were to call in sick to the company while on layover in JFK. After calling HDQ a quick buzz to SOS in order to get a doctor to verify that said Captain does indeed suffer from any number of sympthoms that a medical professional can only diagnose from a patients description yet are serious enough to ground said patient for a number of days. Once management realise that a 777 with 350+ pax cannot take off from JFK with only 2 FOs, talented and qualified and as as they may be, I would imagine that the crewing may change.