The families are looking for answers and redress over the atrocity in their ongoing pursuit for justice.
The victims, who were on their way home, were shot dead near Bessbrook, Co Armagh, after being ambushed by 11 gunmen on 5 January 1976.
They were forced at gunpoint to reveal their religion. They were lined up, and shot dead by the gunmen.
A HET report found the IRA, who claimed to be on ceasefire at the time, was behind what was one of the worst atrocities of the Troubles - under the cover name of the South Armagh Republican Action Force.
The report also revealed that by December 1976, all of the suspects were living in the Republic.
Before travelling to Dublin, families laid wreaths at a memorial to the victims of the massacre in Bessbrook.
Thursday's meeting with the Taoiseach lasted several hours, and relatives were pleased that Enda Kenny listened to their accounts of the atrocity.
Colin Worton, whose brother was killed, said Mr Kenny had listened "intently" to them.
"We're happy with what he gave us, but still we want to follow up now from it," he said.
However the Irish Prime Minister was adamant that he would not apologise for the actions of the IRA.
"I assured them that there is no hierarchy of victims, and that their concerns are every bit as important to me as the concerns of other victims and their families," Mr Kenny said.
"I told them that the IRA was the common enemy of all of the people of Ireland, of all traditions, north and south, and that their campaign of violence was strongly resisted by successive Irish governments."
UUP MLA Danny Kennedy, who was with the families said that an apology was still needed.
We're clear that he's not apologising and nor are we asking him to apologise for the actions of the Provisional IRA but we do believe that there were failings in the political and security system that allowed events such as Kingsmills to take place, and for that, we think that the Taoiseach should make a public acknowledgment and apology.
One man, Alan Black, who was shot 18 times in the massacre, survived by pretending to be dead. He broke his silence after 36 years to tell his personal experience in preparation to meet the Taoiseach along with the victim's families.
He said he wants an apology from the Irish government for failures on their part.
"I think it deserves an apology. You can't defend the indefensible," he said.
The HET found that the weapons used by the gunmen could be linked to up to 100 other killings, including the murders of RUC Chief Superintendent Harold Breen and Superintendent Raymond Buchanan in South Armagh in 1989.
One suspect was also linked to the Omagh bombing.
Colin Worton, whose brother was killed in the massacre, said he wanted to see a public apology as the result of the meeting.
"There are three things I'd like to say and I'd like to see if he does it," he told UTV.
"The first.. is if he gives a public apology, the second one is if he really means it and (is) sincere and the third one is easy, if he is sincere about one and two then he should follow on and hand over the killers of my brother and his nine work mates."
In July the families have demanded a public inquiry in the wake of the HET report into the atrocity.
The Secretary of State and the PSNI said at that time they will not be reviewing the case.