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I think the OP is asking: "without any four of the pieces forming a square."

And, be careful: its sides needn't be parallel to the edges of the board. ;)

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Spoiler

I got 32 this way. No proof it's the most

x x   x x   x  
  x x     x   x
x   x x   x    
      x x     x
    x   x x   x
x         x x  
  x x x x   x x
x   x         x

 

 

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On 3/30/2016 at 11:00 AM, tojo928 said:
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I got 32 this way. No proof it's the most

x x   x x   x  
  x x     x   x
x   x x   x    
      x x     x
    x   x x   x
x         x x  
  x x x x   x x
x   x         x

 

Spoiler

This placement has at least three squares.
So 29 is established as a lower bound:

square_chess_3.jpg

 

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