Is it possible to crash or shutdown the entire Internet?
No. The Internet as a whole is a collection of many independent networks controlled and maintained by different people, businesses, and governments. It has been designed to be redundant, which means even if one portion of the network goes down, users should still be able to gain access through another.
It is still possible for the Internet to experience outages often caused by severe events such as large power outages and earthquakes. When large outages occur, a sizeable portion of the Internet, or even an entire country may be affected. However, even these serious outages will not cause the Internet to shutdown or crash. For example, in early 2007 Asia experienced a series of earthquakes that damaged undersea cables that caused wide Internet and phone related issues in that part of the world. However, the remainder of the world still had access to the Internet.
Also, some governments can setup firewalls and other protection schemes to prevent users from accessing pages on the Internet or totally block access to the Internet. For example, in 2011 the Egypt government shutdown the Internet several times during protests to prevent any information leaking out onto the Internet. However, even during this strict shutdown, citizens were still able to find ways to access the Internet.