Broken square usually refers to an infantry square collapsing or breaking up in battle.
Specific incidents that this expression may refer to are both in the Mahdist War in the Sudan:
- The Black Watch formed one side of a big square under attack, and was ordered by top command to leave that post (leaving their side of that square open) and attack another enemy force which was hidden down a desert gully.
- The restricted rocky irregular ground in that gully made it difficult to form a solid square to resist attack; that square came under intense attack from Sudanese (here, mostly Hadendoa). The square was flooded with a rush of tribesmen and a brutal hand-to-hand fight resulted. The Black Watch were driven back but rallied and eventually drove the Sudanese out, with the square being reformed.
Frank Richards, a soldier in the Royal Welch Fusiliers circa 1901, stated in his memoir entitled 'Soldier Sahib': "If a Welshman went into a pub where a Highland soldier was, of the regiment whose square was once broken by the Mahdi's dervishes in the Sudan, he would sometimes ask for a "pint of broken-square." Then he would have his bellyful of scrapping for the rest of the night, because this was an insult the Highlanders could not forgive." This anecdote is frequently treated as fact.
References
Broken square Wikipedia(Text) CC BY-SA