I had not seen any information on stand-alone builds for Edge Insider on Xbox One.
I assumed since Microsoft was unbundling the browser from Windows, and bringing it to other operating system platforms such as Macintosh and Android, this would happen on Xbox One as well.
Also, I do not know if it would be possible to bring an Edge browser to the Xbox 360 since you plan to deliver Edge to Windows 7 and Windows 8. All of these still use Internet Explorer as the default bundled browser.