The club, who have now resumed training in preparation for the new season which kicks off next month, announced the double deal for the teenagers this week.
It follows the signing on a permanent basis of midfielder Ross Cunningham, from Hamilton Accies.
Midfielder Henderson (18) and forward Jack (19) were both recently promoted to the St Mirren first team squad after featuring last season for the Buddies’ reserve side who reached the fourth round of the Challenge Cup.
Boss Danny Lennon said: “We are pleased to have acquired two players with bags of potential.
“I am very familiar with the high quality coaching of the Saints youth set up. The system there has made great contributions in the development of top-level players like John McGinn and Kenny McLean, whom I gave first team debuts whilst in their teens.
“Jay is a wide midfielder and Josh is a forward who can also play as an attacking midfielder. Jim Goodwin has high hopes for these players, who have joined his first team squad this season and been on the bench for each of St Mirren’s games so far.”
Forward Cunningham made seven appearances during a loan spell from Hamilton last season, before the campaign was cut short.
“We are really delighted to have signed up Ross on a permanent deal,” said Lennon. “We are excited by what he brings. His energy, drive and ability to see things are all tremendous assets.”
Clyde have also signed keeper David Wilson (19), released by St Mirren at the end of last season, as back-up to David Mitchell.