Oxforddictionaries offers this definition:

Having or showing dogged determination not to change one’s attitude or position on something, especially in spite of good arguments or reasons to do so

Several sites have the definition, but not even our reliable Phrase Finder posts an origin.

Word Detective has a related phrase that I’d never heard: “took the studs,” meaning to become stubborn and usually said of a mule. WD found a citation in the Dictionary of American Regional English for “take the studs” (or “get” or “have”) comes from 1797, “and it’s clear that it’s a term primarily applied to balky horses and mules, and, by figurative extension, to uncooperative people.” The word “stud” has an obsolete definition “first appearing in Chaucer in the late 14th century, is ‘defiant of destructive agencies or force; strong, stout.'”

listserv.linguistlist.org has a handful of citations:

• 1771 T. SMOLLETT Humphry Clinker II. 169 The captain..becomes stubborn as a mule, and unmanageable as an elephant unbroke. ·
• 1812 M. EDGEWORTH Absentee xiii, in Tales Fashionable Life VI. 260 She was as obstinate as a mule on that point. ·
• 1853 J. Y. AKERMAN Wiltshire Tales 138 As cam and as obstinate as a mule. ·
• 1922 J. JOYCE Ulysses 411 The likes of her! Stag that one is. Stubborn as a mule! ·
• 1923 Nation (N.Y.) 17 Oct. 432 Then there is the Missouri mule. He it was who won the war. 1972 Listener 21 Dec. 858/2 Not for nothing did the idiom ‘as stubborn as a Missouri mule’ come into the language.

So mules have been identified with stubbornness for a long time. Rinker Buck, in his best-selling book The Oregon Trail, writes that mules are smarter than horses and sometimes smarter than humans who try to drive them into danger. “The common phrase ‘stubborn as a mule,’ [is] a classic example of a man ascribing stupidity to the beast instead of to himself.”