The veteran politician warned the Scottish Government could face years of punishment if it proceeds with the policy, which was the subject of a lengthy court battle.
Speaking at a conference organised by the No to Named Persons group (NO2NP), he also encouraged campaigners to lobby SNP backbenchers to defy ministers over fresh legislation aimed at getting the scheme up and running next year.
Mr Sillars, a vocal critic of the policy, said it would be “politically foolish” for Mr Swinney to press ahead, stating the “logical position is not to proceed”.
Referring to Chancellor Philip Hammond’s Budget u-turn on national insurance, he said: “My advice would be to do a Hammond.
“Last week the government at Westminster was in a hole. It had the brains to stop digging and my advice to this government is this isn’t going to work.
“This is going to be a punishment on you over the years ahead. Stop digging, get out the hole and abandon it.”
Mr Sillars insisted that bringing in the scheme “would sour relations between the SNP government and millions of people”.
“You’re going to 2021, when the next Holyrood election comes,” he said.
“If they force this through, well it doesn’t take long to think of the number of problems that are going to emerge between the statutory bodies, the individual who represents the statutory body and families who fundamentally object to intrusion in their private lives.”
Under the scheme, a single point of contact, such as a teacher or health visitor would be appointed to look out for the welfare of all children up to the age of 18.
The Government was forced to halt its roll-out - due to come into effect across Scotland in August 2016 - after it was challenged in the courts by NO2NP campaigners.
Judges at the UK Supreme Court ruled last year that elements of the policy were “incompatible” with the right to privacy and family life as set out in the European Convention on Human Rights (ECHR).
Mr Swinney announced earlier this month that he will bring forward new legislation to amend the information-sharing aspects of the scheme, in an effort to roll it out in 2018.
The NO2NP campaign wants the policy to be scrapped entirely.
Mr Sillars suggested campaigners should target SNP backbenchers who were elected in 2016 - after the original legislation for the scheme was passed.
He told the conference, held in Edinburgh and attended by around 250 people, that those MSPs may be more “open-minded”, and therefore persuaded to speak out against the party’s “almost Stalin-like discipline”.
“I think they (the backbenchers) have an opportunity to show independent-thinking,” he said.