How much the region you get your PhD is important to your career path?
It's difficult to answer your question with too few details.

You can look at the alumni of the lab/department that gives you the offer. What were their first jobs after leaving the lab? If none of them went to the west, then it is very unlikely that you will do :)

East Asia is very broad, which includes China, Japan, Korea etc... There are only a small set of universities that are known to the west. These are the one or two top universities in the most developed countries in the region, e.g. NUS in Singapore, KAIST and SNU in Korea, Todai in Japan, etc

If you obtain a PhD from a university outside this list, it will be significantly difficult for you to find a position in the west after PhD.

Theoretically, if you have a strong publication records, LoRs from big shots in the fields, you can get a good a position after PhD. But in reality, it is easier to achieve that (papers, LoRs etc) if you are in a top university.