No you don't, but you might as well get your money's worth.
in an emergency situation, they'd almost inevitably put you through a a public patient - at least around here they would - the private hospital here is for non-emergency surgery or for illness that requires hospitalisation for prolonged period. they don't do middle of the night emergency things - it's all done on a booked in basis.

you are under no obligation to use PHI - but in a public hospital, some PHI can help to get a private room so worth mentioning it (esp if you have no gap fee in your coverage)