A boxer who “can’t fight/hit/punch his way out of a paper bag” is a very poor fighter. “From his appearance, one wouldn’t think he was husky enough to punch his way out of a paper bag” was cited in print in 1908. The “way out of a paper bag” saying is now used in other situations besides boxing.
   
[This entry includes information researched by Hugo and published on the American Dialect Society listserv in October 2013.]
   
 
(Oxford English Dictionary)
paper bag, n.
colloq. to be unable to fight (also punch) one’s way out of a paper bag : to be extremely weak or incompetent at fighting. Also in extended use: to be completely ineffectual or inept.
1955   A. Marshall I can jump Puddles (1956) xix. 136   ‘Skeeter couldn’t fight his way out of a paper bag,’ Joe asserted.
1980   Washington Post(Nexis) 14 Feb. (Sports section) f1   Muhammad Ali couldn’t punch his way out of a diplomatic paper bag.
 
24 May 1908, The State (Columbia, SC), “Bloody Britisher Who Wants to Fight Abe Attell,” pg. 18, col. 4:
From his appearance, one wouldn’t think he was husky enough to punch his way out of a paper bag.
(Written by Hype Igoe.—ed.)
 
4 June 1913, Seattle (WA) Daily Times, “Hugh M’‘Intosh says black boxers put him out of business,” pg. 15, col. 1:
“Honest,” continued McIntosh, “Lester could not fight his way out of a paper bag.”
 
30 January 1915, Denver (CO) Post, “Old Swat Belt Happenings,” pg. 6, col. 7:
“Why, Welsh couldn’t punch his way out of a paper bag!”
   
HathiTrust Digital Library
May 1916, Our Navy, “Athletics: Will We Ever Have a Navy Champion?” by “Heinie,” pg. 50, col. 1:
About that time Luther McCarty, who had trimmed them all, looked good, but he collapsed in the ring with Arthur Pelkey. We won’t say Pelkey finished him, because Pelkey can’t hit hard enough to fight his way out of a paper bag.
 
HathiTrust Digital Library
30 July 1921, Collier’s, The National Weekly, “The Lovers’ Handy Man,” pg. 139:
I doubt if Stillwell could punch his way out of a paper bag right now, and this Battlin’ Moore is no cake eater, but a tough boy.
 
Chronicling America
10 January 1922, The Garden Island (Kauai, Hawaii) Sports, pg. 5, col. 2:
Kid Andreas and Bantam Grip on did a brother act in the first. Each refused to strike the other, but seemed content to pose and feint Neither one of them could fight their way out of a paper bag.