I'm confused over which HDHomeRun device I need to purchase for use with my AndroidTV box (Shield).
The HDHomeRun product pages suggest that only the EXTEND supports streaming to phone and tablets, while CONNECT requires that you use your computer. However the HDHomeRun View app suggests that both are supported.
Which one do I need to stream ota hd to my Shield?
You can use either. The difference is that the EXTEND will transcode the TV signal from mpeg2 to mpeg4/264 which is more efficient in terms of size/bitrate (so better experience streaming over wifi) and has wider support in mobile hardware (hw decoding instead of software so less dropped frames).
I believe that the Shield has support for MPEG2 hardware acceleration now so the only question is whether or not your Shield is connected to your network via ethernet. If it is then the CONNECT should work fine. If you can only connect via wifi then you would probably get better, more stable performance from the EXTEND.