Absolutely for any company owned computer and equipment. You should not be using their equipment and work time to engage in any private social activities.
In the case of your private computer and social media, no one should be able to censor your activities (within broad parameters). If, however, your employer does not like what they read or see, it is within their rights to reevaluate your fit for the job with their company. An employer is always assessing the staff to keep the best people for the job and replace the others.
Consider: You work for AA and your facebook pages are littered with pictures of you getting loaded and falling down drunk. Or you work for the police department and there are facebook pictures of your involved in illegal activities.
Just because you can put something on facebook, doesn't mean you should. If the need to express yourself unhindered is more important than your job, then, by all means, carry on. Just expect to pay the consequences.