The Swaziland Federation of Trade Unions and the Swaziland Federation of Labour said their action was a show of "mourning for the death of the judicial and executive systems in Swaziland."
This follows the resignation last weekend of six South African judges who worked on Swaziland's court of appeal.
They resigned after the prime minister said the government would ignore court judgements which curbed the power of King Mswati III.
The judges had ruled that the king was acting unconstitutionally in ruling by decree and overturning court decisions.
Swaziland is Africa's last absolute monarchy and political parties are banned.
Crisis
Earlier it was reported that lawyers had joined judges in going on strike, but it appears they are first asking the prime minister to clarify his contentious comment which was broadcast on TV.
"The prime minister has precipitated a constitutional crisis in Swaziland which can be likened to the overthrow of law and order," Swaziland's Law Society said in a statement.
"To call on law enforcement agencies to disregard the law as pronounced by the highest judicial authority is a promotion of anarchy," it said.
King Mswati, 34, has come under fire recently for insisting on buying a $45m jet at a time when a quarter of the 1m population are suffering from food shortages.
Relations between the royal household and the judiciary have been strained by an unprecedented court case brought by the mother of an 18-year-old girl, allegedly abducted by royal aides to become the king's 10th wife.
Before the case was eventually shelved, the attorney general ordered judges to drop the charges or resign.
'Opposition forces'
On Monday, Chief Justice Stanley Sapire asked in court:
"If government does not honour court rulings, why are we here?"
Prime Minister Sibusiso Dlamini said last week that the judges were being used by "forces opposed" to the monarchy.
Parliament had voted against buying the royal jet but the government said it would still proceed with the purchase.
The government said the jet would help him travel the world, asking for food aid.