What is the maximum and minimum value for x?

The maximum value of angle x will be 45°, putting E and F on the line between A and C.

The minimum value of angle x will be arctan(1/3), about 18.4°, when E and F are on lines BC and AD, respectively, and located 1/3 of the way along those segments.

Update:

The maximum value of x will be the complement of the minimum value, about 71.6°.

Update2:

The locus of points E or F will be on the black circle in the diagram. The tangents from corner A have angle x values of π/12 and 5π/12 (green arcs). This is a slightly larger range of angles than those obtained where the black circle intersects the edges of the square, i.e. at a distance 1/3 along the side.

The derivation is not difficult. It involves writing an expression for the distance of a point (say, E) from A and from the center of the square. The former distance will be twice the latter. The equation can be transmuted into one for the black circle, which has its center 2/3 along the diagonal from A to C. The radius is obviously √2 times 1/3 the diagonal. The tangent to the black circle is found on the blue circle, which has the same radius and is also centered on the diagonal. Since the center of each circle is on the other, you will probably recognize that from either circle center, the short arc between intersecting points is 120°, so the angle between the two red lines is 60°.