Why, in A Midsummer Night's Dream, does "square" mean "quarrel"?

The sense of to "fall out, to be at variance or discord, to disagree or quarrel" (OED, sense 8b of square, v.; paywalled) is obsolete. It appears in historical dictionaries. OED attests the sense with quotes from 1542 to 1608.

We actually still have versions of this usage in modern English: to square up to or against¹ someone means (from Cambridge Dictionaries)

to prepare to fight, compete, or argue with someone:
The players squared up to each other and started shouting.

And to square off means (also Cambridge)

to oppose someone in a competition or prepare to fight someone:
Bradley is expected to square off with Cook in the next election.

And a square go is a Scottish term for (Collins Dictionary)

a fair fight between two individuals

This "fighting" usage of square seems to be related to the idea of facing one's opponent squarely (still Cambridge) and squaring one's shoulders (Merriam-Webster) when readying oneself for an unpleasant task, as well as other square-related concepts that have to do with forthrightness and not shirking, such as foursquare (M-W), on the square, and the other ways in which one can square up (both Collins). It may also be related to the idea of two individuals being at right angles to one another.²

The Oxford English Dictionary¹ dates most of these positive senses of the word to the late 16th through the 17th century. They seem to have arisen metaphorically from the square's geometric traits: a square is upright (as opposed to any shape with non-right-angles), shows its full face (since a square seen at an angle doesn't look like a square), and is equitably arranged (since all its sides and angles are equal).

So Oberon and Titania were squaring in the sense that they were coming face-to-face with one another, and not edging away or sloping off from the fight.

¹_{My impression is that the to version is more British (square up to is actually listed as a UK usage in several dictionaries) and against is more American (I hear squared up against especially in US sports contexts).}

²_{Compare the adjective cross and the verb thwart (both OED, see note below), which both evolved from the idea of being at a right angle (to something) to, respectively, being generally angry and actively opposing someone.}

³_{Note that the OED Online is unfortunately a prescription service (behind a paywall). However, many folks, especially in the UK, will have access through their local public library, and most students should be able to access through their institution's library. More info available at About the OED, including, as of February 2019, a special subscription price that is available for one more month in celebration of the OED's 90th anniversary.}