For any shape that is not constant width, there are many different maximal elements of the same diameter containing it.
Constant width shapes are maximal. They are regions swept out by line segments of length $D$ swept by their midpoints moving perpendicularly along curves perpendicular to along smooth curves (with cusps) whose tangent line turns 180 degrees
Any shape that is not constant width has some projections that do not have maximal diameter. For each projection that has maximal diameter, there is a diameter as above perpendicular to the line of projection.
Any method of interpolating a positively turning curve between the existing diameters will create a maximal element of the set of shapes of the given diameter. There are always infinitely many ways to do this unless the set is already maximal --- any one solution for an interpolating curve can be perturbed anywhere it is not constrained.
This observation can be applied, for exmaple, to the case of squares mentioned in the question. For example, you can begin enlargement of a square by adding an arc of a circle centered at one corner and passing through an opposite corner, then take the convex hull. More systematically: the square only defines two diameters. There are many Legendrian sections of tangent line bundle of plane $\to \mathbb {RP|^2}$ (cf. Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?) interpolating these two elements.
The total area is locally given by an easy formula, but globally it seems like a highly irregular function, since it depends on how the sweeping diagonal overlaps itself. I doubt if there is a good theory of a maximal area shape of the same area containing the given one.