Good news, one NeoPixel stick works :). Following the basic instructions found on Yono's LEDscape repository, combined with a 74HCT245N and I've got flashing NeoPixels run off my BBB. The important part is making sure to connect the 74HCT245 correctly, otherwise nothing will happen. Make sure that the BBB is on the Ax side and the NeoPixels are on the Bx side, and make sure DIR is +5V and /OE is 0V. Everything *should* just work after that.
You may have luck with a different level shifter, but the 74HCT245 seems to be highly recommended by a bunch of people, so I went with that.
My next step is going to be trying to implement some Python code to do the logic in my program, and use Yono's LEDscape code just to drive the NeoPixels.