My view is that HTML is just one way of declaring the DOM, but ultimately HTML or some enhanced version of it is just a string. The virtual DOM is another path to constructing the DOM, but instead of a string it's javascript upfront.
Flux is not a framework so until there is a clear way to use it beyond a single page of short examples it's not ready.
React is also only the view where Ember and Angular are the full MVC. If you have a single page app that has maybe one view with a lot of components maybe React makes sense. Anything beyond that though and it absolutely does not.
Need a login page? a profile page? etc.. All that stuff will be separate React applications served from separate HTTP endpoints which is going backwards IMHO.
That said, you can absolutely have a multi-page SPA built with React. Check out React-router.
The code that communicates with the backend should be in your Flux Actions.
To check if you're authenticated before rendering a view, intercept the router transitions.