The name Dongguk (東國) comes from the Chinese words for east (東) and realm (國). It's a term from Buddhist cosmology: the eastern paradise,* which is ruled by Aksobhya Buddha. Dongguk University, with branches in Seoul and Gyeongju, takes its name from this cosmological term, and as you might guess, the university is itself Buddhist.
I'm applying there. My application to Sungkyunkwan fell through yet again, and I've heard nothing from Hanyang, but it would be a hoot to end up at a Buddhist university after having taught at a Catholic one. Then again, I'm also applying for work at Catholic University in Seoul, although I'm not sure Catholic will allow me to roll over my E-1 visa. I've sent CU an email to ask about the visa situation; we'll see.
Meanwhile, I'm hanging my hopes on finding refuge in the eastern paradise.
*Often called the Eastern Pure Land, given the tradition that cleaves to this cosmology.