Does Korea count for overseas service bars?

No Overseas Service Bar for Korea, unless you meet these time period and criteria.
(2) Korea, between 27 June 1950 and 27 July 1954. Credit toward an overseas service bar is authorized for each month of active Federal service as a member of the U.S. Army serving in the designated hostile fire area in Korea between 1 April 1968 and 31 August 1973. The months of arrival to, and departure from the hostile fire pay area are counted as whole months. If a Soldier receives a month of hostile fire pay for a period(s) of service in Korea, then the Soldier may also receive credit for a corresponding month towards award of an overseas service bar.
Maybe you are referring to the KDSM and OSR.

May 31, 2016 to June 7, 2017? If that's the case then yes you should have an overseas ribbon on your ASUs. Check your ERB also

The stripes you were talking about is combat stripes, not overseas. You get a ribbon for overseas after a certain time serving overseas. Then you get numerical numbers put on that ribbon if you've gone back overseas. Also, you need orders for them.