Can I assign my students to view internet videos outside of class, from sources like YouTube or NPR?
Unless otherwise stated on the website in question, the answer is yes. Sites like YouTube, NPR, CNN, most major news sites, etc. put video content up with the intent of sharing it. Moreover, many of these sites will include "embed codes" that allow for the direct linking of such videos in course management systems like Blackboard.
See this YouTube embed code as one such example:
<iframe width="560" height="315" src="http://www.youtube.com/embed/339ZGwWQVXM" frameborder="0" allowfullscreen></iframe>
Note that I used this code to embed the below YouTube video to this answer. Also note that this is a PBS video posted to YouTube, with the specific intent of sharing.