Green’s Dictionary of Slang

button-down adj.

also buttoned-down
[the button-down collared shirts from Brooks Brothers (New York) that are the uniform of the US business establishment]

(US) conforming, holding establishment, conservative values.

[US]S. Bellow Henderson The Rain King 309: He was no cold-storage, Ivy League, button-down, broken-hipped civilian like you.
[US]H. Ellison ‘Memory of a Muted Trumpet’ in Gentleman Junkie (1961) 107: One of the button-down boobs let out a [...] whistle.
[US]C. McFadden Serial 32: A superstraight with a button-down banker’s mind.
[US]S. Morgan Homeboy 4: She’d fallen in [...] love with the buttondown square of the millennium.
[US]J. Stahl Permanent Midnight 66: A tall, buttoned down man who’d started out penning Newharts.
[UK]Guardian Rev. 3 Dec. 2: Baxter was the quintessential button-down schnook.
B. Woodward (ref. to 1966-1972) Secret Man 193: [N]ot all FBI men are alike. Some agents are well defended within their official persona and play to the stereotype—buttoned-down, tight and by-the-book.