Q: Device isn't eligibile for requested build ??

Upgrade it to what? You show to already be running 8.4.1.

Sorry, but since you agreed to a Non Disclosure Agreement in the Public Beta that cannot be discussed in a public forum like this. Go back and make sure you read the instructions completely and that you downloaded and installed the profile required to update.

If you see Error 3194, Error 17, or 'This device isn't eligible for the requested build' in iTunes - Apple Support

redundant post so deleted - already answered by previous posters

No, it doesn't mean that. It means on the primary computer you use with your phone you should check the host file section as they recommend and edit as necessary.

Have you previously jailbroken or rolled back the OS on your iPhone? Have you run a non-authorized beta on this phone previously? It does sound like it with the error message you are getting. Go back to the beta page and follow the directions closely on how to roll back your phone to 8.4.1. If you do not understand what is on that page you should NOT be attempting to use the public beta.

So it could have been jailbroken. What I would do is check the computer you use and make sure the host files are correct as outlined in that article. Then I would put the phone in DFU mode (Google it for helpful videos) and Restore it in iTunes as a new device.

If you put it into DFU mode, restored it as a new phone and still received an error 3004 there is a problem with that phone, return it and get your money back.