Let $s(n)$ denote the sum of digits in the binary expansion of the integer $n$. Hare, Laishram and Stoll (2011) studied the number of odd integers such that $s(n)=s(n^2)=k$, for a given integer $k\geq 1$. The remaining cases that could not be treated by theses authors were $k\in\{9,10,11,14,15\}$. In this paper we show that there is only a finite number of solutions for $k\in\{9,10,11\}$ and comment on the difficulties to settle the two remaining cases $k\in\{14,15\}$. A related problem is to study the solutions of $s(n^2)=4$ for odd integers. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that $n=13,15,47,111$ are the only solutions. In this paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for $s(n^2)=5$.