The origin and meaning of “(Caught) on the horns of a dilemma” turned out to be more difficult than I imagined. The meaning was the easier part, and the origin more difficult.
Meaning : Refers to a situation where one is confronted with making a decision based on two options, the results or consequence of either decision having equally unpleasant results. So, no matter what decision you have to take, the outcome is unfavourable.
From the 1933 print of the Oxford Shorter Dictionary, Dilemma has Greek origins. Di meaning two and lemma an assumption or a premiss. According to the same source dilemma was a form of argument,(Rhetoric), in which an adversary was given the choice of usually 2, "but possibly more" alternatives equally unfavourable. The alternatives are the horns of the dilemma, either of which can cause pain.
The origin of the horns I could not determine. Most refer to the horns of a bull, and even to the devils horns. Somewhere I saw a source that quoted a Greek translation meaning a four legged animal with a tail, but lost the site.
An example of the phrase in modern terms by Oxford " the dilemma of a swimmer among drowning men, who all catch at him" (Page 510).