Can your manager make you take down a union flyer? Only if the company prohibits postings of any kind.
If employees are allowed to post any non-work material (Seattle Reign poster, seasonal holiday decorations, Black Lives Matter sign, Pride flag, etc) in their work areas or in non-work areas such as break rooms, then you can post a flyer supporting a union. If management tells you to remove a union flyer but not a Seahawks Super Bowl banner, that’s disparate treatment. Contact WPEA staff or stewards.
The freedom to organize and form a union is protected by the National Labor Review Board (NLRB) and the 1st amendment to the U.S. Constitution. In addition, WPEA Higher Ed contract Article 2.2 specifically prohibits workplace discrimination based on an employee’s participation or non-participation in union activities.
Want to read more? Check out the Washington State Workers Rights Manual