No, because that would defeat the purpose of having public healthcare: a universal safety net for those who for some reason lose out on private healthcare. But

No, because that would defeat the purpose of having public healthcare: a universal safety net for those who for some reason lose out on private healthcare. But because private healthcare has to compete with public healthcare in countries with such a dual system, the price for private insurance is driven very low, so it isn't as disastrously expensive as in the United States, for example, and demanding a large tax refund for this would seem extreme.