Green’s Dictionary of Slang

1. to act like a coward, to shirk one’s duties.

1882		Wkly Graphic (Kirksville, MO) 18 Aug. 2/1: If Arthur and his crew have not ‘flinked’ and gone back on the decision, then we are mistaken.
1887		E. Custer Tenting on the Plains (rev. edn 1895) 388: All the boys done bully, but Corporal Jackson—he flinked. The way he flinked was, to wait till the boys had drove the Injuns two miles, and then he hollered, ‘Gin it to ’em!’.

2. to play truant from school.

1969

M. Kantor Missouri Bittersweet 152: ‘I’m not going to school. I’m going to flink.’ You remember that used to be slang for not going to school? [...] My son and daughter tell me nowadays that it’s merely ‘cutting classes’ [DARE].

3. (UK gay) of a male homosexual, to have a relationship with a woman in order to appear heterosexual.

1972

B. Rodgers Queens’ Vernacular 30: to date a woman to prevent suspicion of being homosexual [...] flinking (Brit gay sl).