This afternoon, my SO stepped outside and heard them bleating, so he went out to track them down. I saw him head towards the barn, so I assumed that's where he heard them, so I went back to watching TV. After a while I began to become concerned that it was taking him a long time to return, so I went outside to check on him.
He was outside the fence, in the woods surrounding the creek. The goats had indeed gotten outside and were hiding in a tree! He had managed to catch one of the goats, Josephine, which he handed to me through the fence. I dropped her inside the barnyard fence (which has smaller holes, definitely too small for them to fit), and we set out trying to catch the other goat.
Fortunately the two are sisters and don't like being separated. I ran around the barnyard, the barn, through the front gate, and back around to the woods so I could help, but by then Buttercup had managed to squeeze back through the outer fence and was standing outside the barnyard fence opposite her sister. I ran all the way back around, then snuck up on Buttercup and grabbed her just as she started to run off.
I dropped the goat inside the barnyard fence with her sister and then nearly passed out. I'd been running around in the sun for nearly a half hour, with the heat index at least 120°F (49°C). My SO had been in the woods, so he was spared some of the heat, but he ended up completely coated in mud from falling in the creek, and he was covered with scratches, bug bites and nettle stings.
But the goats are back, locked up in the barnyard which they (hopefully) will find harder to escape from. They'll have to stay there until they grow big enough they can't escape through the outer fence. They'll also spend time with the sheep, whom we're hoping will adopt them.
The goats appear to be in perfect health; there was plenty of fresh, running water in the creek, shade in the woods, they could climb trees to escape predators, and there was plenty of greenery for them to eat (not to mention the neighbor's corn fields). It was all just a grand adventure for them.