Like could care less, give a damn is a Negative Polarity Item, that is, a phrase that is ordinarily used only within the scope of semantic negation of some kind (not, never, only, rarely, few, etc.). Hence the perceived strangeness of They could give a damn, which has no overt negative, but means the same thing as the same phrase with a negative. I.e, the business manager was saying that his members couldn't give a damn.
Give a damn is a member of the open Minimal Direct Object class of NPI's, like lift a finger, drink a drop, do a thing, eat a bite, etc. The implication of all of them is that, if one can't even Verb a Minimal Direct Object, why, then, one couldn't Verb any Direct Object at all. Thus it's an idiomatic intensification of a negative. But it does usually require a negative to intensify.
However, there apparently is such a thing as negation by association. Like what happened to French pas from ne...pas, which is now usable as a negative in its own right, from long association in the discontinuous morpheme with the overt negative ne, give a damn and could care less have, in American usage at least, come to have their own quasi-independent negative force.
Give a damn has been used independently of negatives for at least 25 years in America. I published a paper (J. Lawler, Ample Negatives, in Papers from the Tenth Regional Meeting, Chicago Linguistic Society (CLS 10)) in 1974 that remarked on this topic, among other negative phenomena.
