"We were eyeball-to-eyeball and the other fellow just blinked"

This quote by Dean Rusk is a metaphor for a circumstance in which 2 celebrations are in a fight and one of them backs down. It suggests that a person party wanted to take a stand and the other was not. This could be interpreted as a sign of strength and guts on the part of the celebration that did not blink, as they wanted to stand their ground and not pull back. It might likewise be analyzed as an indication of weak point on the part of the celebration that blinked, as they were not willing to take a stand and instead picked to back down. This quote might be used to describe a range of scenarios, from a political dispute to a personal dispute. Ultimately, it is a suggestion that sometimes it is required to take a stand and not back down, even when the other party hesitates to do so.