I've had not much hope to find one in an acceptable time but after running another couple of hours my program delivered this square-free end-configuration for n=12:
oooooooxxxox
xoxoxoxoxoxx
xoxxooxooxxo
oooxxxooxxoo
oxooxoxxxoxo
xxxooxoxoxox
xoxxxxooooxx
oxoxoxxoxoox
ooxoooxxoxox
xxooxooxoxxo
oooxoxoxooxx
xoxxxoxxxoox
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CSI of Charles University
Drawn game for n=12
I've had not much hope to find one in an acceptable time but after running another couple of hours my program delivered this square-free end-configuration for n=12:
oooooooxxxox
xoxoxoxoxoxx
xoxxooxooxxo
oooxxxooxxoo
oxooxoxxxoxo
xxxooxoxoxox
xoxxxxooooxx
oxoxoxxoxoox
ooxoooxxoxox
xxooxooxoxxo
oooxoxoxooxx
xoxxxoxxxoox