Jumping all over the place

You are allowed to put a checker on any lattice point in the Cartesian plane with $y$ -coordinate less than or equal to $0$ (i.e. a point with integer coordinates on or below the $x$ -axis). Note that you may place infinitely many checkers to start.

The only legal moves are horizontal or vertical jumping--a checker can leap over a neighbor, ending up $2$ units up, down, right, or left of its original position, provided that the destination point remains unoccupied. After the jump is complete, the checker that was jumped over is removed.

What is the minimum positive integral $y$ -coordinate that a checker cannot attain?