While we don't have a native app for the Fire, you may be able to launch the HipChat the web app inside the Silk browser. To do so, open up the browser and log into the HipChat web UI and click the 'launch the web app' button. Alternatively, try appending /chat to your HipChat URL. Such as:
What I'm not sure of is how the Silk browser will handle loading the web app. You may also be able to look on the Fire app store for any third-party XMPP clients. These apps should be compatible with HipChat since it's XMPP compliant. You can find your XMPP info by going to https://mycompany.hipchat.com/account/xmpp
Hope this helps!
HipChat Server Support Engineer | Atlassian
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