Answer
For Integer 22 the given statement is not true.
Work Step by Step
At most 2 squares means no squares, 1 square or 2 squares or you can say 0 squares, 1 squares or 2 squares.
Let us choose the integer 22, the squares that the numbers can obtain are $1, 4, 9, 16$ (Since the terms in the non-negative terms in a sum cannot exceed the sum).
First case No squares
22 is not the cube of the integer.
Second case 1 square
16+6=22 ⇒ 6 is not the cube of an integer
9+13=22 ⇒ 13 is not the cube of an integer
4+18=22 ⇒ 18 is not the cube of an integer
1+21=22 ⇒ 21 is not the cube of an integer
Third case 2 square
16+9=25 >22 Not possible
$16+4+2=22 \Rightarrow 2$ is not the cube of an integer
$16+1+5=22 \Rightarrow 5$ is not the cube of an integer
$9+4+9=22 \Rightarrow 9$ is not the cube of an integer
$9+1+12=22 \Rightarrow 12$ is not the cube of an integer
$4+1+17=22 \Rightarrow 17$ is not the cube of an integer
Conclusion: the statement is not true for 22 and thus we have disproved the statement.
