Jay L. Halio notes in his New Cambridge Shakespeare edition of King Lear (Cambridge University Press, 1994) that the meaning of "the most precious square of sense" is uncertain and that various interpretations have been proposed. He provides an example from The Riverside Shakespeare:
Riverside glosses 'square of sense' as figurative for 'the human body' or 'human life' and cites [The Faerie Queene] II, ix, 22. In Pythagorean terms, the square is an emblem of the material world, or the world of sense, the physical universe; the circle, an emblem of the conceptual world, even God himself. (…)
G. K. Hunter's New Penguin Shakespeare edition (Penguin, 1972) glosses the line as follows:
which the sense, not 'out of square' (unbalanced) but in their proper constitution — the constitution that is so precious to us — possess
Nick de Somgyi's edition King Lear: The Tragedie of King Lear (The Shakespeare Folios. Nick Hern Books, 2004) uses "professes" instead of "possesses". "Possesses" is the word found in the First Folio (1623), whereas the 1608 quarto edition used "professes". Most modern editions reject "professes" because Regan used "profess(e)" just two lines earlier and treat "professes" as a typesetter's error (an 'eye-skip').
Both the 1608 quarto text and the 1623 First Folio contain the words "most precious square of sense"; de Somogyi gives several examples of interpretations that have been provided for these words:
The four-square of the 'nobler senses' (sight, hearing, taste, smell) (William Warburton, 1747); the comprehensive input of all five senses (Thomas Edwards, 1748) — but why, therefore 'square'? (…) John Bulloch ingeniously suggested that QF's 'square of sense' was a misreading of 'quintessence'. Other editors preferred to offer a range of suitably abstract paraphrase: 'the finest susceptibility, or the highest capacity of happiness' (Henry N. Hudson, 1851-6); 'the most delicately sensitive part of my nature' (W. Aldis Wright, 1877, recorded by H. H. Furness, 1880); 'the choicest symmetry of reason, the most normal and intelligent mode of thinking' (Alexander Schmidt, 1879). (…)
de Somogyi concludes,
Perhaps these nuances are, finally, unwanted: Regan's hyperbolic claims are, after all, both insincere and meaningless, however weighty (or 'ponderous', as Cordelia calls them) they may seem.