Is that a tick attached underneath the beak?
When I looked at the photos at home I saw that one of the two was carrying a bug in its beak and it seemed to be holding on to it, not passing it on to the other during the entire time that I was watching. They were totally oblivious to me following them with my camera.
Does it look like a doubled-over ant?
.... or perhaps a beetle?
The behavior seemed very odd. So I checked the article on Brown Creepers on BNA Online and found this as a possible explanation:
High-pitched call given, followed by a silent chase in which pair, about 1–3 m apart, spiraled around tree trunks with white undersides prominent, and landed on a tree trunk, one above the other. Wing-fluttering, rapid beating of wings held above the body, often occurred after a chase. This display often followed by another chase. Courtship feeding often occurred after these displays and continued throughout the nesting cycle until eggs hatched, male usually feeding female. Female wing-flutters and gapes similar to the begging of young birds; male lands above her, turns sideways, and places food item in female’s throat. Copulation not observed during these displays. Authors: