COCA (Corpus of Contemporary American English, 1990-2011) has
11 occurrences of stubborn as a mule and
none for obstinate as a mule.
BNC (British National Corpus, 1980s-1993) has
3 occurrences of stubborn as a mule and
none for obstinate as a mule.
BUT the COHA (Corpus of Historical American English 1810s-2000) has
20 occurrences of stubborn as a mule and
19 occurrences of obstinate as a mule. Here's an example:
Never would take my advice. Obstinate as a mule. But I'll pay her out yet, ha, ha! Forgery! Scandal, ha, ha! (Houghton Townley, The Scarlet Feather, 1909)
Seems to confirm what other people have said. I think you can still say
obstinate as a mule, but it sounds more literary.