Hint: what point or points on the square are the farthest from its center, and if those points can lie anywhere in space, what 2D shape do we call the set of all points equidistant from a given center?
What do you mean by a narrow entrance?
Is this whole scenario in 2D space? Then if I understand your question correctly it would just be 1m since its the shortest possible length i.e. it would just be whatever the length of the side of square is.
by
I think this is similar to the moving sofa problem except without the long corridor.
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.