Interesting - a couple of my colleagues suffer the same nosebleed problem when working 777 flights (we operate 772ER).
Can the pilots control the airpacks on the 777? If so, I wonder if they are turning them off or reducing the flow? I haven't suffered any problems on the 777 except once when I woke up from crew rest feeling terrible and so did the rest of the crew. I'm pretty sure I remember a discussion with the flight crew revealing they did 'something' to the air flow.