Can my employer prevent me from wearing more than one (eight?) pro union button?

Yes.  The Second Circuit just ruled that a "one button" rule did not violate the NLRA.  Employees at Starbucks wanted to wear multiple pro union buttons.  Generally, wearing pro union symbols is permitted by the National Labor Relations Act.  But, there is a limit.  The Second Circuit wrote:

Starbucks is clearly entitled to oblige its employees to wear buttons promoting its products, and the information contained on those buttons is just as much a part of Starbucks's public image as any other aspect of its dress code.   But the company is also entitled to avoid the distraction from its messages that a number of union buttons would risk.   The record reveals that one employee attempted to display eight union pins on her pants, shirts, hat, and apron.   Wearing such a large number of union buttons would risk serious dilution of the information contained on Starbucks's buttons, and the company has a "legitimate, recognized managerial interest[ ]" in preventing its employees from doing so.   The company adequately maintains the opportunity to display pro-union sentiment by permitting one, but only one, union button on workplace clothing.   Starbucks has met its burden of establishing that the one button restriction is a necessary and appropriate means of protecting its legitimate managerial interest in displaying a particular public image through the messages contained on employee buttons.

So, next time you are at Starbucks, check out the buttons.  If you see more than one pro union button, warn the employee that their job could be in jeopardy.