chuck (n.1)

"piece of wood," 1670s; "piece of meat," 1723; probably a variant of chock (n.) "block." "Chock and chuck appear to have been originally variants of the same word, which are now somewhat differentiated" [OED].

Chock and Chuck, Are low terms, very frequently used before full,--as the coach was chock full of passengers. The house was chuck full. [Daniel Powers, "A Grammar on an Entirely New System," West Brookfield, 1845]

Specifically of shoulder meat from early 18c. Meaning "device for holding work in a lathe or other machine" is from 1703 (also chock). American English chuck wagon (1880) is from the meat sense. 

chuck (n.2)

"slight blow under the chin," 1610s, from chuck (v.1). Meaning "a toss, a throw" is from 1862.

chuck (v.1)

"to throw," 1590s, variant of chock "give a blow under the chin" (1580s), possibly from French choquer "to shock, strike against," imitative (see shock (n.1)). Meaning "pat playfully, give ablow to" is from 1610s. Related: Chucked; chucking.