Square board means that side a = side b. Open means that he doesn't necessarily need to finish where he started.
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No - in particular, you can easily see that it isn't possible on a 2x2 or 3x3 board.
This paper might interest you. It mentions Schwenk's theorem, which allows you to determine whether a knight's tour is possible very easily from the dimensions of the board (which may be rectangular, in general):
http://faculty.olin.edu/~sadams/DM/ktpap…
The theorem is this:
An m x n chessboard with m ≤ n has a knight’s tour unless one or more of these three conditions holds:
(i) m and n are both odd;
(ii) m = 1, 2, or 4; or
(iii) m = 3 and n = 4, 6, or 8.
This paper might interest you. It mentions Schwenk's theorem, which allows you to determine whether a knight's tour is possible very easily from the dimensions of the board (which may be rectangular, in general):
http://faculty.olin.edu/~sadams/DM/ktpap…
The theorem is this:
An m x n chessboard with m ≤ n has a knight’s tour unless one or more of these three conditions holds:
(i) m and n are both odd;
(ii) m = 1, 2, or 4; or
(iii) m = 3 and n = 4, 6, or 8.
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Definitely not every square board. On a 2x2 board you can't move at all. On a 3x3 board you can't move to the middle square if you start at the edge and you can't move to the edge if you start in the middle. If you restrict the size to be at least 4x4 I don't know the answer.