"to vomit, eject the contents of the stomach," 1600, probably of imitative origin (compare German spucken "to spit," Latin spuere; also see spew (v.)). First attested in the "Seven Ages of Man" speech in Shakespeare's "As You Like It." Related: Puked; puking.
1737, "a medicine which excites vomiting;" by 1938 as "material thrown up in vomiting," from puke (v.). U.S. colloquial meaning "native of Missouri" (1835) might be a different word, of unknown origin.
It is well known, that the inhabitants of the several western States are called by certain nicknames. Those of Michigan are called wolverines; of Indiana, hooshers; of Illinois, suckers; of Ohio, buckeyes; of Kentucky, corn-crackers; of Missouri pukes, &c. To call a person by his right nickname, is always taken in good part, and gives no offence; but nothing is more offensive than to mis-nickname—that is, were you to call a hoosher a wolverine, his blood would be up in a moment, and he would immediately show fight. [A.A. Parker, "Trip to the West and Texas," Concord, N.H., 1835]
Bartlett (1859) has "A nickname for a native of Missouri" as the second sense of puke (n.), the first being "A mean, contemptible fellow." The association of the state nickname with the "vomit" word is from at least 1858, and folk etymology talks of the old state literally vomiting forth immigrants to California.
late 14c., "act of expelling contents of the stomach through the mouth," from Anglo-French vomit, Old French vomite, from Latin vomitus, from vomitare "to vomit often," frequentative of vomere "to puke, spew forth, discharge," from PIE root *weme- "to spit, vomit" (source also of Greek emein "to vomit," emetikos "provoking sickness;" Sanskrit vamati "he vomits;" Avestan vam- "to spit;" Lithuanian vemti "to vomit," Old Norse væma "seasickness"). In reference to the matter so ejected, it is attested from late 14c.
"brownish-purple," literally "flea-color," 1787, from French puce "flea-color; flea," from Latin pucilem (nominative pulex) "flea," from PIE *plou- "flea" (source also of Sanskrit plusih, Greek psylla, Old Church Slavonic blucha, Lithuanian blusa, Armenian lu "flea").
[T]he couleur de Puce, or flea colour, and the couleur de Noix, or nut colour, are the reigning winter taste. [Westminster Magazine, January 1777]
Perhaps so called as the color of the scab or stain that marked a flea-bite; flea-bitten was a color word in English to describe whiter or gray spotted over with dark-reddish spots (by 1620s, often of the skins of horses, dogs, etc.). That it could be generally recognized as a color seems a testimony to our ancestors' intimacy with vermin.
OED sees no connection between this word and obsolete puke (16c.-18c.; hence Shakespeare's puke-stocking) as the name of a dark color of now-uncertain shade (Century Dictionary says perhaps reddish-brown, OED says bluish-black or inky; others suggest grey).