"The Mining Town Four" was Dr. James Corenthal's pet name for the four children that he adopted from the Fairmount Children's Home: Vincent, Evan, Jeffrey and Stephanie.
In a record from the Children's home Corenthal details the intake of Evan, a controversial patient also referred to as HABIT A second record details the intake of Vincent to the facility, and refers to both Evan and Vincent as having come from a "town of ash," presumed to be Centralia, Pennsylvania. The details of how Jeffrey and Stephanie came to the Fairmount Children's Home have not been revealed.
The first reference to the name "Mining Town Four" is found a letter from Corenthal, where it was revealed that Dr. Corenthal and his wife Maryann had adopted the four children and were raising them. In the same letter he refers to the last of the children as having "succumbed." The details of what that meant was unclear until one of the articles in the Microfilm Collection (retrieved during the video One step forward, two steps back) revealed that the children had died in the same timeframe mentioned here. Also in the microfilm was an account of Dr. Corenthal getting into a fight with some locals while “looking into the history of [his] children’s former lives." He told the police that "The locals simply didn’t want anything to do with me.”
Beyond sharing names, the relationship between The Mining Town Four and the cast of EMH is unclear at this point, though it is clear that there is a connection. This connection is part of the basis of the Iteration Theory.
In the video "Lexi," HABIT explains to Vinnie that there are people involved in this who are "on the inside" and some who are "on the outside." Vinnie and his friends (presumably HABIT means the rest of the cast,) are on the "inside." This status could be a reference to being part of the Mining Town Four.