Green’s Dictionary of Slang

cube n.2

[an intensified version or ‘superlative’ of square n. (4b)]

an extreme conservative, an ultra-respectable person.

D. Burley N.Y. Amsterdam Star-News 27 Sept. 15: The kid is such a ‘solid square’ that I’ve coined the name ‘Cube’ for that variety.
[US]Lait & Mortimer USA Confidential 186: When the cubes are rolling in the county area the most popular places are Wolhurst.
[US] ‘Bop Dict.’ in Mad mag. Mar. 20: Cube – a 3-D square.
[US]W. Brown Teen-Age Mafia 52: I see you know how to handle the stuff [i.e. marijuana], baby. Maybe you ain’t the cube I thought you was.
[UK]R.A. Norton Through Beatnik Eyeballs 26: Up till now I been a made-up cube.
[US]T. Wolfe Electric Kool-Aid Acid Test (1969) 408: The New Dimensions finish their set and they’re mad as hell, of course. What . . . cube was doing that screaming bit, f’r chrissake.
[US]T.M. Kochman ‘The Kinetic Element in Black Idiom’ in Kochman Rappin’ and Stylin’ Out 164: Earlier terms such as ‘square’ and ‘cube’ also reinforced the settled and immobile quality of the persons so designated.

In derivatives

Cubesville (n.) [-ville sfx1 ]

the fig. ‘world’ or mindset of ultra-conservative, highly respectable people.

[UK]J. Osborne World of Paul Slickey Act II: He’s strictly from Cubesville.
‘Yulesville’ in Bench Racer at r.webring.com 🌐 His threads were from Cubesville and I had to chuckle, / In front, not in back, was his Ivy league buckle!
cubistic (adj.) [SE sfx -istic]

(US) extremely conventional.

[US]Young Savages [film script] Mother, you can be such a drag sometimes, so utterly cubistic.