Divide the square into four isosceles triangles, with the square's center as the apex of each of the triangles and the sides of the square as the bases.

As a coordinate system, choose the center of the bottom edge of the square as the origin. The right edge of the bottom triangle is then y = 1/2 - x. The point on this line that is also on the boundary of the area to be calculated has sqrt(2) * x = y.

sqrt(2) * x = 1/2 - x

x * (1 + sqrt(2)) = 1/2

x = 1 / (2 * (1 + sqrt(2)))

y = 1/2 - 1 / (2 * (1 + sqrt(2)))

The area sought is a central square plus four segments of parabolas, as the boundary is a set of four parabolas with the center of the square as a focus and each of the sides of the square as directrices. The central square then has each side equal

2 * (1/2 - (1/2 - 1 / (2 * (1 + sqrt(2)))))

= 2 * 1 / (2 * (1 + sqrt(2)))

= 1 / (1 + sqrt(2)) = sqrt(2) - 1 =~ 0.414213562373095

This is also the base of the piece of parabola included in one of the constituent triangles. The height of that piece is

1/4 - (sqrt(2) - 1) / 2 =~  0.0428932188134524

The area of a parbola cut off by a chord is 

(2/3) * b * h

In this case thats

(2/3) * (sqrt(2) - 1) * (1/4 - (sqrt(2) - 1) / 2)

which simplifies to 

5/(3*sqrt(2)) - 7/6 =~ 0.0118446353109123.

The area sought is the total of the central square plus four of these parabola segments: