Accreditation of high schools is managed at the provincial rather than federal level, which means each province manages its own educational system.
Public high schools are funded and offer quality courses developed by the provincial government. Independent or private schools receive both public and private funds and teach courses approved by the government. Some provinces only regulate what diplomas a school is accredited to award, rather than whether a school is accredited to operate, and you should make sure that the private school you are interested in can issue the diploma you wish to receive.
You can also look to see if a school is accredited by any independent accrediting bodies, such as the Canadian Educational Standards Institute (CESI), or has memberships in organizations like the Canadian Association of Independent Schools, the Quebec Association of Independent Schools, etc. There are also accrediting bodies associated with particular religions or educational philosophies. As an international student looking to study in Canada, it is important to make sure that the school you choose can give you the education and graduating diploma you want.