I have TVEverywhere/Channel Plus installed on my QNAP TS-253D NAS. On my desktop on the same LAN I have the Hauppauge Win-TV-quadHD. Can I use the desktop turner card to capture over-the-air shows that are no longer available on the TvEverywhere DVR?
No. The only hardware tuners directly supported are SiliconDust HDHomeRun tuners. You can try using something like Tvheadend as a middleware to try to get them to work, but such configurations are not directly supported.