Based SOLELY on the information in your post, yes, they can be fired for not signing it.
However, the devil is in the details, and if you want a carved in stone answer, you will have to take a copy of the document in question to an attorney in your state. The exact wording matters and it is possible (not likely but possible) that something in the wording may offer protection.
It is slightly more likely that some or all of the document may be unenforceable. But, while possible, it is VERY unlikely that anything in the document creates job protection.