Let's divide the square into four equal parts.
And its side be 2x

Now, area of each square = 4x²
Radius of each circle = x
Area of circle inscribed in it = πr² = πx²

Now, the area of remaining part in the square other than circle
= 4x² - πx² = 4x² - 22/7x²

The area in remaining part of the square is equal to the part in b/w the four circles.

22/7 cm² = 4x² - 22/7x²
22/7 = 22/7 x²
x = 1 cm

So, the radius of each circle is 1 cm
Since the circles touch each other as well as the square, the side of the square will be the sum of diameters of two circles= 2d=4r
The radius of the circle is r cm
The area of the square - area of the four circles= 24/7
 (4r)^{2} -4π r^{2} =24/7
16 r^{2} 
-4π r^{2} 24/7
4 r^{2} (4-π)=24/7
4 r^{2}  r^{2} 6/7
7/6(4 r^{2}  r^{2} )=1
14/3 r^{2} - 11/3 r^{2} =1
 r^{2} =1

Therefore the radius of the circles is 1 cm..

Hope it helped you!!  :)