The delivery of the boxes, containing essential items such as clothes, books and blankets, will begin on Tuesday.
The national roll-out follows pilot projects in Orkney and Clackmannanshire which started in January.
The boxes, inspired by Finnish baby boxes, can also be used for babies to sleep in.
New parents are being urged to consider safe sleeping practices for their newborns ahead of the launch.
Boxes issued under the Scottish scheme meet the highest safety standards, and have been awarded British Safety Standard accreditation as a crib for domestic use, the Scottish Government said.
Safe sleep guidance is provided in a leaflet inside the box and safety guidelines are also printed on the box lid.
Chief medical officer for Scotland, Dr Catherine Calderwood said: “All parents should think about sleeping practices at the earliest possible stage even before their baby is born.
“Early evaluation shows the scheme is helping raise awareness of safe sleeping practices and we are committed to evaluating how the baby box is helping families across Scotland.
“It is important parents make the safe sleeping choice that is right for them, and anyone needing more information should speak to their health visitor or midwife.”
Lynsay Allan, executive director of the Scottish Cot Death Trust, said: “Whilst the proven safest place for a baby to sleep is in a cot, crib or moses basket, I also recognise that the box can be used as a safe sleep space for newborn babies, particularly where families may have no alternative and where they might otherwise unsafely share their bed or sleep on a sofa or chair with their baby.”
The reassurances over the use of the boxes for sleeping follows concerns raised by cot death charity The Lullaby Trust.
Earlier this month, chief executive Francine Bates, said: “We support all efforts to promote safer sleep for babies, however we do have concerns about the baby boxes being marketed as products which will reduce infant mortality and SIDS (sudden infant death syndrome).
“It is also not possible for baby boxes to meet all current safety standards, as nursery furniture regulations only apply to traditional cots, cribs and bassinets, not boxes made from cardboard.
“If parents choose to use the box to sleep their baby, we urge them to read and follow our advice, approved by our scientific and paediatric advisers.”