No, not on this board. A 5 by 5 grid has 25 boxes, an odd number, so one player always claims more boxes than the other.
No. All 25 boxes are always claimed by the time the 60th line is drawn, and 25 cannot be split evenly between two players. One of you finishes ahead by at least one box.
Close games
The closest possible result is 13 boxes to 12. Because closing a box earns another turn, a single long run at the end can swing a game that looked decided, so a lead is not safe until the last line is drawn.
See how big the board is for why the count is odd, and the strategy guide for how to win the closing stretch.