The task is
Number the grid 1 to 25, from left to right top to bottom. Also, color the grid like a checkerboard so cell 1 is black. Pressing a white cell only changes black cells, and vice versa.
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
There are an odd number of black cells which all must be changed. Pressing an interior white cell changes an even number of black cells, while a white cell on the border changes an odd number of black cells. This means that the total number of border white cell presses must be odd.
Note that cells 2 and 6 are the only ones which affect cell 1. This means cells 2 and 6 combined are pressed an odd number of times. Same goes for the other pairs of white cells bordering the other corners, (4,10), (16,22) and (20,24). Combining these four observations, the number of times a border white cell was pressed must be odd + odd + odd + odd = even, contradicting the first paragraph. Therefore, the puzzle is unsolvable.
This method does not generalize to an n x n grid. I've written a program (using Gaussian elimination over the finite field of order 2) to determine if an n x n grid is solvable, and it appears to be solvable exactly when n is even.