Barry McElduff has now deleted the tweet, which was posted shortly after midnight on 5 January, and issued an apology.
Ten Protestant workmen were murdered by the IRA in the Kingsmills massacre on 5 January 1976.
Mr McElduff said he had no intention to offend, but unionists have condemned his actions.
DUP MLA for Newry & Armagh William Irwin has called on the Sinn Féin leadership to explain why Mr McElduff posted “such a bizarre video”.
“Was it tomfoolery or a blatant insult to innocent victims? The timing and brand of bread raises many questions about the motivation behind this video,” Mr Irwin said.
“Given Sinn Féin’s repeated insensitivity to victims and glorification of terrorists, PIRA victims throughout the United Kingdom have interpreted this video as a calculated and deliberate insult.
“Regardless of what explanation Mr McElduff or his party would attempt to offer, they should apologise to the families of those murdered at Kingsmills.”
DUP leader Arlene Foster tweeted at lunchtime today: “Kingsmill victims were shot by the IRA after being asked their religion.
“Shame on any elected rep who posted that inhuman video.I feel sorry for IRA victims & West Tyrone who have McElduff as their MP.
“All murder was wrong.
“Glorifying any murderer is sickening.
“Mocking is depraved.”
Ulster Unionist councillor David Taylor said Mr McElduff should be ashamed of himself.
“I pose this question to Mr McElduff and to Sinn Féin – where is the respect shown to the families of the victims of the Kingsmills Massacre with disgusting actions like this?” he said.
“This provides clear evidence of Sinn Féin’s real attitude towards the victims of republican violence and towards unionism in general.”
His party colleague, councillor Chris Smyth, said he would be reporting Mr McElduff to the Commons Standards Committee and to the PSNI.
Traditional Unionist Voice leader Jim Allister said: “It stretches the bounds of credibility well beyond breaking point to suggest that it is anything other than deliberate that this video was posted on the same day as the 42nd anniversary of the Kingsmills massacre.
“Sinn Féin’s utter contempt for victims is clearly on display here. Any talk of equality or human rights from that party is once again exposed as so much cant and hypocrisy.”
Alliance Party leader Naomi Long tweeted to Mr McElduff: “I see you have deleted your video. Will you also explain what on earth you were thinking of, posting this on the anniversary of the Kingsmills Massacre?
“Have you any apology to make to those victims and survivors deeply hurt by your antics whether deliberate or not?”
In his apology, Mr McElduff said: “Had not realised or imagined for a second any possible link between product brand name and Kingsmills anniversary.
“Further, I apologise for any hurt or offence caused. Never my intention to offend anyone who has suffered grievously.”
In a later statement, he added: “I apologise unreservedly for the hurt and pain this post has caused.”
The 10 workmen were shot dead on 5 January 1976 after the gunmen stopped their van and asked which among them was a Catholic, and instructed that man to leave the scene.
At the time it was claimed by a group calling itself the South Armagh Republican Action Force – a cover name used by the Provisional IRA.
The Historical Enquiries Team of the PSNI later stated that the attack was carried out by Provisional IRA members.