Insurance companies, by and large, make their money on the delta between the prices they charge and the costs to deliver service. If they can delivery service to a predominately healthier population, then they can deliver service for lower costs, meaning they can lower costs to be more competitive, while being able to make more profits and deliver better outcomes.
The notion that they want to drive up healthcare costs is kind of ridiculous. If that were true, they'd be rushing to sign up patients with terminal illnesses and costly pre-existing conditions, but the fact of the matter is that the government had to literally force them to for that to happen.
Here are a some links I've collected on the topic - obviously, I'm a little biased but I have yet to see any compelling case for privatised healthcare on essential services:
http://www.canberratimes.com.au/comment/private-health-insur...
http://www.abc.net.au/news/2017-02-23/more-health-insurers-t...
https://www.theguardian.com/australia-news/2016/nov/18/exper...
https://www.theguardian.com/australia-news/2016/nov/17/marke...
https://www.theage.com.au/politics/federal/crisis-looms-as-a...
http://www.smh.com.au/money/saving/dont-be-fooled-by-the-pri...
http://www.abc.net.au/worldtoday/content/2015/s4262363.htm
http://www.smh.com.au/comment/why-you-dont-need-private-heal...