The only way

⌊x⌋² + ⌊y⌋² = 1

can be true is for values of (x,y) such that 
either 

⌊x⌋² = 0 and ⌊y⌋² = 1 or 
⌊y⌋² = 0 and ⌊x⌋² = 1 or

⌊x⌋² = 0 when and only when  0 ≦ x < 1
⌊x⌋² = 1 when and only when -1 ≦ x ≦ 1
⌊y⌋² = 0 when and only when  0 ≦ y < 1
⌊y⌋² = 1 when and only when -1 ≦ y ≦ 1


⌊x⌋² = 0 and ⌊y⌋² = 1 when and only when 0 ≦ x < 1 and -1 ≦ y ≦ 1
⌊y⌋² = 0 and ⌊x⌋² = 1 when and only when 0 ≦ y < 1 and -1 ≦ x ≦ 1
 
Thus it is true at all points interior and on the square excpt at the
four corners



Its area is 2x2 or 4 square units.

Edwin