What does Boxing Day have to do with boxing?
Old English bog "shoulder, arm," extended in Old English to "twig, branch" (cf. limb (n.1)), from Proto-Germanic *bogaz (cf. Old Norse bogr "shoulder," Old High German buog, German Bug "shoulder, hock, joint"), from PIE *bhagus "elbow, forearm" (cf. Sanskrit bahus "arm," Armenian bazuk, Greek pakhys "forearm"). The "limb of a tree" sense is peculiar to English.