Maybe this is too ongoing to answer, but anyway...
There seems to be travel alerts for (South, at least) Korea due to MERS at the moment.
Is Jeju Island Affected As Well?
What Will Summmer Travel Be Like?
While I can answer this question as a native South Korean, this really is a question you should ask your country's relevant government office.
As of June 15th, 2015, there are no known cases of MERS infection in Jeju Island. But as this is an on-going event, the situation can change at any minute.
Most countries issued a level 1 travel advisory or equivalent regarding MERS outbreak in South Korea. As long as you take precautions such as washing your hands often, you'll be safe. While MERS is definitely a serious issue here, the media has blown it out of proportion as if this disease will be the demise of the country. It hasn't changed the lifestyles of locals here, apart from maybe seeing some people wear masks.
For others going elsewhere than Jeju, do avoid hospitals with MERS patients in it.
Above all, keep an eye out for your travel advisories issued by your country.