Yes and no. It depends on whether the lawyers for the insurer can satisfy the court by putting forward evidence that production of this information is relevant.
There are numerous cases in which an order for production of eg. private Facebook pages, has been given, but in these cases there was evidence to show that the information was likely to be relevant to the issues in the claim.
There are also cases where the courts have rejected a request for production of social media. So if an injury client admits to having a Facebook account and that family members post photographs to the Facebook page, but the publicly-available photographs show nothing special in terms of activities and no strenuous activities, the court will not easily reach the conclusion that the private photographs would be any different.
The lesson is, if you have been injured in an accident and want to take a claim, make sure to discuss the issue of use of social media with an experienced injury lawyer. In Newfoundland and Labrador, Ches Crosbie Barristers is ready to help.