Who all is using an Nvidia SHIELD here? I'm streaming from my PC to my 4k TV for some games, specifically The Witcher 3 and For Honor. Both are more enjoyable relaxing with a controller in my opinion. Here's my setup:
PC: i7-4790k + 1080 Ti, connected to wireless AC router (~800mbps local connection)
SHIELD: Connected to router via CAT6 cable
TV: Samsung UN60KS8000
 
Generally no problems with stuttering or lag. Pretty impressed, honestly.
 
One complaint: I still haven't figured out how to get a headset working with the SHIELD. Audio streams from the PC but haven't been able to get the mic to stream back to the PC. Seems like a big thing to miss in this development...

I own the original SHIELD.  I am not sure what you are talking about because it sounds vastly different.  I wish NVIDIA didn't call everything SHIELD and instead came up with an original name.
 
What I own:

 
What NVIDIA later released with the same "SHIELD" name:

 
What you own? Maybe?:

I've now seen some artifacting on the SHIELD TV unfortunately. Little grey squares that jump around on the screen when in game streaming from my PC. 
Time to open up a discussion with Nvidia...

It sounds like you are streaming your game from your computer to your TV.  What you are seeing sounds like completely normal video compression.  That sort of thing happens when you stream a game as a video between two devices and need to receive and display it very quickly in order to reduce input lag.
 
I haven't streamed any games on my SHIELD since I am completely opposed to the idea of streaming games.  Why not just move your computer to your TV or move your TV to your computer?  It is very inefficient to stream a game and handicaps the experience in one way or another.
 
 
Try this:
Set your SHIELD resolution to 1080P (if possible).  Let the TV do the upscaling (if possible).  Then, lower SHIELD FPS setting from 60 FPS to 30 FPS or Auto.  Lower FPS should increase image quality.  But... will lower the FPS (of course).  We're back to the original problem of streaming a game between devices, aren't we?  
 
 
See also:
https://forums.geforce.com/default/topic/840710/shield-tv/shield-tv-gamestream-compression-artifacts-are-extremely-visible-even-on-wired-gigabit-and-it-look/post/4761511/#4761511