Your new boss might be insecure and trying to assert her authority. Talking about coworkers to each other was definitely unprofessional.
But, that doesn't excuse any unprofessional behavior on your part either. Many companies (my day job included) block many websites, and the company is justified in asking you to do company work on company time. If you need a break, you could try a trip to refill your water bottle, use the restroom, stand and stretch, etc.