UAW President Bob King wrote a two-page letter to a reporter explaining why he wasn't allowed to park his Toyota Camry at a UAW hall lot (or any union property): Toyota hates the First Amendment. [KCBJ, Detroit News]
And Coca-Cola requires any company working with them to not drink Pepsi products for the duration of the contract. My companies lawyers weren't huge fans of that clause as they would come after you for breach of contract if you were seen violating it. (basically on coke grounds, in the office, or any where you would be in contact with the team that could recognize you)
Microsoft asks that iPods and iPhones be left outside of their offices.