Say the doorway is d meters across.

At some point in time, the center of the square has to cross the threshold of the doorway. At that point, it must be at a distance e <= d/2 meters from one of the sides of the doorway. Thus the center of the square must be at a distance at most e meters from some point outside the square or on its boundary. But the closest point on the boundary of the square is 1/2 a meter from the center. So d/2 >= e > 1/2, and therefore d > 1.

Conversely, if d > 1, it's easy to fit the sqaure through by pushing it head-on through the doorway.

So the answer is any doorway wider than 1 meter.