No, that is not always true. Medical care is definitely an exception to that rule. Medicine and medical care is not a luxury, and is one of the least elastic things in the market. When needed people will pay anything to get medical care. Insurance is not a traditional product that is subject to supply and demand.
One again this would only be true is insurance worked like a product, it does not. Insurance works a lot more like a slot machine. The cost of playing a slot machines do not go down when a new casino is built, they just pay off less often.
They might bitch about it but they would pay it anyway, because medicine is not an elastic product.
Lets remember also that while we are all bitching about rising premiums insurance companies are not increasing their rates because they are losing money. Across the board all the large insurance companies are posting record breaking profits. They are raising premiums because the market will pay it. Because in real world market a thing is worth whatever people will pay for it, and right now we will pay for higher premium insurance. What is the Republican plans going to do about that?