This question touches on the political history of Northern Ireland, which is long, complicated and contentious, but the following is my best understanding.
The "union" in this case is the one created in the Acts of Union of 1800 between the Kingdom of Great Britain (itself created by a union between England and Scotland) and the Kingdom of Ireland. This act has been repealed in the Republic of Ireland, but still applies in Northern Ireland, forming the sovereign state of the United Kingdom of Great Britain and Northern Ireland. Safeguarding here means preventing its dissolution and repeal, here presumably through a stealth, 'little by little' tactic in which the Republic of Ireland would gradually increase its control over the rest of the island.
In general, the DUP have opposed any change of the situation on the ground, which could be seen as putting forward the concept of a single Irish nationality in the island of Ireland (including in 2018), or of anything signalling a move away from a constitutional position that Northern Ireland was now and always an integral part of the UK. They also expressed dissatisfaction with the concept of governing with Sinn Féin, whose leadership at the time included people accused or convicted of terrorist offences, and at a time when the IRA was still armed and active (although on a 'ceasefire').