Yes, it is absolutely true. Literally from the Canada Health Act: "Private health insurance plans are prohibited from duplicating coverage for health services provided in Canada which are insured under the Canada Health Act."
If you start to provide services normally covered by the government, you will be shut down, or you'll have to take your case to the Supreme Court where this law is still being tested.
There are places that provide parallel services, they operate in a grey area. For example, the Supreme Court of Quebec ruled that private services can be provided for treatments wherein the government does not provide 'timely service' i.e. 'wait times are too long'. But exactly the parameters of those 'wait times' nobody knows, and the only way to find out is to go to the Supreme Court. So not a good business plan.
This is not at all true in Canada. I'm really confused by this statement.