Indeed, given any solution in which some squares remain empty, we can add circles to some of the squares maintaining the equality of the number of filled circles and open circles remaining.

PROOF:
Each row or column which has an empty square in it, has at least 2 empty squares - since the grid is 10 by 10, and an even number of squares are filled in each row/column.

Let us choose a row r_1 that has an empty square, and then choose a column c_1 which meets r_1 at an empty square. Then choose a row r_2 that is not r_1 but meets c_1 at an empty square. Similarly, let c_2 be not c_1 but meet r_2 at an empty square. Continue.. Let r_n be a row that is not r_{n-1} but meets c_{n-1} at an empty square, and let c_n be a row that is not c_{n-1} but meets r_{n-1} at an empty square.

We can continue doing this as long as we wish, as each row/column with an empty square has at least two empty squares. Eventually we will repeat. Suppose that r_i=r_j is the first repeat in the rows with i