Further solutions were submitted by Joe Thornton, Steven Goggins, Eric Baldwin, Courtney Pelowski, Kristin Hellem.  Further solutions were submitted by Bryan Fluher, Al Zimmermann, Burkart Venzke, Nancy Schwarzkopf, Scott Powell, Sudipta Das, Lou Cairoli, Paul Botham, Bill Webb.

We can partition the first quadrant into "bad" and "good" regions.  The bad region consists of the points in any of the regions (n,n+1] ´ (n,n+1], where n is any non-negative integer.  The region is depicted in purple on the left.  The good region consists of everything else.  If the initial point lies in the bad region, you let your opponent start, otherwise you start.

The reason is the following:  The bottom left-most square of the bad region is a losing postion.  Anyone who finds him or herself there has lost the game since any further move will send you out of the allowable region.  Any player in the good region will be able to force the opponent into the bad region in one move, whereas any player finding him or herself in the bad region will be forced to move into the good region in one move.

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