You do not have to call in weekly. If you choose not to call to make a statement, that's your prerogative.
If you are mistakenly terminated, it is the employee that will have to fight to get reinstated. Once an employee is terminated it's nearly impossible to fully rectify the damage done.
Benefits stop, pension contributions stop, seniority is interrupted, permanent record will reflect termination. The records at the union, the company, and all of the benefit providers will be affected.
When you show up at the pharmacy or doctor and find that your prescription card and benefits have been canceled or 10 years from now when you learn that your seniority was interrupted you may question whether it was such a good decision.
I know, you can file a grievance, you can take it the the panel, you can do what ever you want. The people involved on the management side have no incentive to help you and even if they choose to help fix the issues it's not within their direct control.
Refusing to be cooperative can be a two way street. There's probably at least one management person out there that's prerogative would be to make your life difficult just to make a statement.