No need for the "right" to work. CX is a foreign airline with foreign registered aircraft aircraft with pilots with foreign licenses overseen by a foreign regulator. If you look carefully on the CX tail, you won't see any stars and stripes.
For simplicity, the pilots will based in HKG. They will do some regional flying but will also have a "preferred rostering destination." Days off will be given down route. i.e. in JFK. Local national pilots will go home and foreign pilots will travel to their home country on their days off thereby keeping immigration happy since they won't be overstaying their crew visas (183 days).