In a written statement the group said they were concerned the conflict would "become a gradual ethnic cleansing".
It is not clear whether the six are demanding political asylum, but they have said they will stay inside the embassy "until we get a better deal".
Violence in the Moluccas is reported to have claimed another 30 lives in recent days, following continued clashes between rival Christian and Muslim gangs.
Since the first clashes in January last year more than 3,000 people are thought to have been killed.
The six Moluccan Christians say they want a UN force to be deployed to the islands along the lines of the operation in East Timor.
'Ethnic cleansing'
In a written statement the group said they were concerned the conflict would "become a gradual ethnic cleansing".
"We want the world to pay attention to the Moluccas because fighting is still rampant in the Christian and Muslim quarters," one supporter is quoted as saying.
There has been no comment so far from Swiss officials.
The six, who include one woman, said they had chosen the Swiss Embassy for their actions because the UN commission on human rights is headquartered in Geneva.
Civil emergency
Recent reports from the Moluccas say the state of civil emergency declared in the islands has done little to bring the fighting to an end.
The latest violence appears to be focussed on the island of Ambon where local residents say women and young children are among those to have been attacked.
The government has sent reinforcements to the region, but there have been repeated accusations of military officers either taking part in the clashes or standing by and refusing to intervene.