mire (n.)

the meanings "mass of small, cryptogamous, herbaceous plants growing together" and "bog, peat-bog" are the same word: Old English meos "moss plant" and mos "bog;" both are from Proto-Germanic *musan (source also of Old High German mios, Danish mos, German Moos), also in part from Old Norse mosi "moss, bog," and Medieval Latin mossa "moss," from the same Germanic source.

These are from PIE *meus- "damp," with derivatives referring to swamps and swamp vegetation (source also of Latin muscus "moss," Lithuanian mūsai "mold, mildew," Old Church Slavonic muchu "moss"). The Germanic languages have the word in both senses, which is natural because moss is the characteristic plant of boggy places. It is impossible to say which sense is original. The proverb that a rolling stone gathers no moss is suggested from 14c.:

Selden Moseþ þe Marbelston þat men ofte treden. ["Piers Plowman," 1362]

Moss-agate "agate stone with moss-like dendrite forms (caused by metallic oxides)" is from 1790. Scott (1805) revived 17c. moss-trooper "freebooter infesting Scottish border marshes" (compare bog-trotter).

"abounding with mud, swampy, boggy," late 14c., from mire (n.) + -y (2). Related: Miriness.

1570s, "soft, wet, boggy land; a marsh," from obsolete quag "bog, marsh" + mire (n.). Early spellings or related forms include quamyre (1550s), quabmire (1590s), quadmire (c. 1600), quavemire (1520s), qualmire.

The extended sense of "difficult situation, inescapable bad position" is recorded by 1766; but this seems to have been not in common use in much of 19c. (absent in "Century Dictionary," 1897, which does, however, have a verb, marked "rare," meaning "to entangle or sink in or as in a quagmire"). It revived in a narrower sense in American English in reference to stalled military actions, 1965, with reference to the U.S. war in Vietnam (popularized in the book title "The Making of a Quagmire" by David Halberstam).