This is false. Consider the number $7$.
Work Step by Step
Consider the number $7$. The only squares smaller than it are 1 and 4. It is impossible to add $3$ or fewer of these and get $7$. To see this, first note that there must be exactly $1$ $4$ -- if there are more, the sum will be too high, and if there are less, the sum will be too low. Then, the highest that can be madei s $4+1+1 = 6$, so it is impossible and the statement is proven false.