The four sides of the square are directrices of parabolas with the origin as focus, with arcs of these parabolas separating the area of interest from the rest of the square.

For simplicity lets take the bottom one of these arcs and lift it by 1 unit: y = .5 + a*x^2 such that, for example,

.5 + a/4 = sqrt(1/4 + (.5 - a/4)^2)

Wolfram Alpha solves this as a=1/2, so

y = (1 + x^2) / 2

We need to take this to its intersection with y = 1-x. This solves to x = sqrt(2) - 1. One of the areas we need to subtract from the area of the square is

integral{1-sqrt(2), sqrt(2)-1} (1+x^2)/2

 = 2/3 (4 sqrt(2) - 5)

 

 Four of these must be subtracted from 4, as well as four times four small corner squares of side equal to 2-sqrt(2):

 

 4 - 4*(2/3 * (4*sqrt(2)-5) + (2-sqrt(2))^2)

 

 which simplifies to (4/3) (4 sqrt(2) - 5)

 

 This is about 0.87580566598984, which sounds about right, as its somewhat smaller than a central unit square.