Recently my company has instituted a policy making it mandatory to clean the break room, which some of the employees do not use(smokers), due to some people who do use it basically trashing the area. When we refused to clean the break based upon the fact that we do not use this area and that as smokers we are required to maintain our outside areas while other employees are not required to help us clean it, we were then written up and/or suspended. The suspended employees had refused to sign a write up.

My first question is can they legally make a policy like this and not enforce that other employees also have to clean the smokers area?

Second can they legally suspend an employee for not signing a write up that they do not agree there is basis for?

Yes and yes.