It seems that until recently, Dropbox was not available via the Chrome OS file manager. In order to use it you would need to access the Web interface. However, due to some API changes and coding by GitHub user
To set it up, install the File System for Dropbox and run it. On first run, you’ll be asked to provide your Dropbox credentials, and mount the drive. From that point forward, you should see Dropbox in the file manager.
First, you as you move down through folders, they will not be listed on the left, only at the top. (In the screenshot I’m in /dropbox/Camera Uplaods, but I do not see that folder structure on the left as you do with Google Drive.)
Second, I seem to be unable to download content directly into Dropbox. Every attempt I have made to select a Dropbox folder as a Save As location has ended up as a failed download from the browser. Instead, I just save to the downloads folder then move the file later.