According to the Ministry of Justice, effective April 1, 2010, you will have to show your health insurance card--either kenko kokumin hoken (National Health Insurance) or shakai hoken (Company Health Insurance)--when you apply to renew your visa or change your visa status. If you do not belong to one of these plans, you will be encouraged (forced?) to join or your visa could possibly be revoked.
How do you get coverage? You can apply for kenko kokumin hoken at your ward office; you can apply for shakai hoken through your employer.
It'll be interesting to see how the eikaiwas react to this. For decades, they have avoided enrolling their employees into the system by playing with the number of hours worked per week, a subject that was brought up here back in 2005..