I live and teach at a university in S. Korea. I may have an opportunity to complete my PhD in TESOL or English linguistics at this university. However, I am reluctant because I have heard that PhDs from non-Western universities (in particular, Korea) are not valued or equatable to US PhDs. Anecdotal evidence from my department supports a similar notion: all the PhDs in my department were received in the US. So, my question is, how are PhDs from non-Western universities perceived? Would a university rather hire someone with a US PhD rather than a US citizen with a Korean PhD?
I may have an opportunity to complete my PhD in TESOL or English linguistics at this university
If you decide to do PhD in English linguistics, I would advise you to do it in an English speaking country. The reason is simple, how do you convince others that PhD in English linguistics in S. Korea, where English is a foreign language, is worth something?
On the other hand, if you decide to do PhD in TESOL (Teaching English to Speakers of Other Languages), where that "other language" is Korean, then I think a Korean university may be a reasonable choice if the quality of both the PhD program and the university are quite good.
Your job opportunity in Korea after you obtain your PhD may not be good if you stay in Korea after you graduate. However, you may actually have advantage over somebody who receives PhD from a Western university because their lack of practical knowledge - teaching English to Koreans if you go to countries other than Korea.
I met many Chinese/Koreans in the US, who had gone to English schools. Many of them told me they wish their English teachers had trainings in their native languages so they would have less difficulties while learning English. I deeply believe those English teachers should have gone through trainings by the PhDs like you.
Would a university rather hire someone with a US PhD rather than a US citizen with a Korean PhD?
Presumably you mean a US university and not a Korean university. The school from which you obtain your PhD has some impact on the hiring process at US universities, but not much. Having a PhD from a top rated department with a "famous" advisor can be beneficial, but a PhD with an advisor from a university (whether it is a US university or a Korean university) that the search committee isn't familiar with is pretty common. For research intensive universities the most important factor is your future ability to publish and secure external funding.