If you are on the horns of a dilemma, you have to choose between two things, both of which are unpleasant or difficult.
The bird is caught on the horns of a dilemma. Should it attack the predator, even though it then risks its own life? Or should it get out while the going is good?
I’m really on the horns of a dilemma here - do I say no to this great job opportunity, or do I accept it and move away from my family?