Entry in progress—B.P.
 
     
Dictionary.com
box·ball  /ˈbɒksˌbɔl/ [boks-bawl]
–noun a game played between two players on two adjoining squares or sections of a sidewalk or a playground, in which a ball is hit back and forth between the players, each defending a square, the object being to prevent a fair ball from bouncing twice before hitting it back into the opponent’s square. 
[Origin: box1 + ball1]
Based on the Random House Unabridged Dictionary, © Random House, Inc. 2006.
 
(Dictionary of American Regional English)
box ball n chiefly NYC
An outdoor ball game; see quot. 1937.
1937 (1947) Bancroft Games 477, Box ball..differs from Hand Tennis only in the ball used and the absence of net and foul lines…The server stands behind his end line and bounces a handball, driving it with a stroke of the open hand so that it goes over the line into the opponent’s court. The receiver..drives it back with the open hand, either after it has bounced or..on the fly.
1968 DARE (Qu. EE33,..Outdoor games) Inf NY 119, Box ball.
1968 DARE Tape NY118, There was also box ball, which was played only with two boxes, sort of like tennis or ping-pong but without a net but with boxes.
1975 DARE File NYC (as of c1925), Box ball..a form of handball in which the players faced each other in a game similar to tennis—but without a net—and hit a rubber ball back and forth in a “court” formed by the more-or-less standard 4-foot-square concrete pavement outlines, the seam between them serving as the “net” beyond which the player must hit the ball.
1977 New York Times (NY) 6 July 29/3 NYC, Box ball—street tennis, using the much-loved pink rubber “spaldeen,”—is still box ball and 21 points still win.
 
29 August 1972, New York (NY) Times, “Spaldeen Olympiad Is Proclaimed—and Anyone Can Play” by Michael T. Kaufman, pg. 35:
The events include stickball, stoopball, Chinese handball, boxball and Johnny on a Pony. All but the last are played with a pink rubber ball, known as a Spaldeen.
(...)
BOXBALL. Played by two players on two adjoining boxes on the sidewalk. Also played with a Spaldeen. Game is like tennis, Ping-Pong, badminton, with players slapping the ball over center line between boxes. Twenty points wins.