On Monday, The News Letter obtained a copy of an email sent to health chiefs by the Department of Health's permanent secretary, Richard Pengelly.
He said he was concerned about the "overtly political tweets from colleagues across the service".
He added that it was "particularly relevant during the election".
The permanent secretary, who is married to DUP MLA Emma Little Pengelly, said "such public statements" were "not appropriate".
Northern Ireland goes to the polls on 2 March to elect 90 members to the Northern Ireland Assembly.
The trust's email said: "At all times, but particularly now as we are in the midst of the pre-election 'purdah' period, staff should be reminded that as public servants, we must at all times remain publicly neutral in terms of expressing any political opinions for example, through social media.
"The trust would take this opportunity to remind you of your responsibilities".
The trust then sent a follow up email to outline the section of the NI Executive's Election Guidance  which is relevant to Non-Departmental Public Bodies.
The Department of Health said that while staff are entitled to express personal opinions on social media, these must be "in line with the HSC Code of Conduct".
Employment lawyer Rosemary Connolly from Warrenpoint, said staff receiving such an email was "highly unusual".
"Individuals are entitled to have political opinions and most will have. They are entitled to express them.
"The difficulty would arise if someone was expressing an offensive, aggressive opinion or harassing someone.
"But simply expressing a political viewpoint in a tweet is a lawful activity, therefore it seems to me difficult to understand how an employer could require no employee tweet on any political matter.
"Unless there was a provision in the contract, or a social media policy, and I would be surprised if that was the case, then people are entitled to these views. They're only subject to normal rules - defamation or breaching confidentiality in some way.
"If you're on social media in your private time, you're entitled to freedom of expression as anyone would would be, providing it doesn't contravene some other obvious situation. "
A Department of Health spokesperson said: "HSC staff have the right to express their own personal opinions on social media.
"However, this should be in line with the HSC Code of Conduct, applicable to all HSC employees, which sets out the core standards of conduct expected of HSC staff and which complements individual organisations policies and procedures."
Dr Brian Patterson, former chair of the British Medical Association in Northern Ireland, told the BBC that he thought the email was very unusual.
"If these are people's personal accounts this action is somewhat unusual, even extreme," he said.
"While I was chair of the BMA, I was always careful about what I said in public but that was because I represented the BMA and other doctors."
He said people were entitled to their own views especially in Northern Ireland where "health is a very prominent issue".
Royal College of Nursing Northern Ireland Director Janice Smyth said it "actively encourages members to engage with the political process".
"We believe that this is an important way in which nurses can influence change and help shape health and social care policy," she said.
"In doing so, they should be mindful of their Code of Professional Conduct and associated professional guidance on the use of social media, ensuring that their personal communications reflect appropriately upon the nursing profession.
"Provided that these principles are adhered to, the RCN sees no valid objection to nurses expressing their personal political views on personal social media sites or, indeed, in any other form of personal communication."
Staff in the South Eastern Trust have been sent an email instructing them not to express political opinions on social media.