Down syndrome is named after John Langdon Down, a British physician who first fully described the characteristics of the syndrome in 1866. Some countries, most notably the U.K. and Australia, refer to it as Down's syndrome. In the U.S. that would denote Dr. Down as having had the syndrome. It is an area of confusion and argument, but either way is fine.
What is more important is the use of person-first language. Individuals with Down syndrome are people first - they are not "downs", "downs kids", or "downs people".