I should have read the referenced article by Bacher and Eliahou before trying to find drawn games. But I've read it now.
Among many other things, the authors describe parametric families of square-free configurations for n=14.
Not all of those configurations do provide the required relation of the numbers of symbols of both sorts needed to be an end-configuration of the game.
But one can, for example, take the family A1, give the variables x1 up to x8 the value 0 (or O, resp.) and the variables x9 up to x16 the value 1 (or X, resp.).
Then the numbers of symbols of both sorts are equal and that square-free configuration is also a correct end-configuration of the game for n=14.
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On drawn games for n up to 14
I should have read the referenced article by Bacher and Eliahou before trying to find drawn games. But I've read it now.
Among many other things, the authors describe parametric families of square-free configurations for n=14.
Not all of those configurations do provide the required relation of the numbers of symbols of both sorts needed to be an end-configuration of the game.
But one can, for example, take the family A1, give the variables x1 up to x8 the value 0 (or O, resp.) and the variables x9 up to x16 the value 1 (or X, resp.).
Then the numbers of symbols of both sorts are equal and that square-free configuration is also a correct end-configuration of the game for n=14.