Here is the full list of all 28 Square Route achievements.
Complete the "Tutorial" levels
Complete the "No Problem" levels
Complete the "Troublesome" levels
Complete the "Devious" levels
Complete the "Rock Hard" levels
Complete the "What The?!" levels
Use bricks 1000 times
Climb 250 ladders
Use gravity 500 times
Use turntables 250 times
Use springs 250 times
Dig 1000 times
Crumble 1000 broken platforms
Pick up 1000 extra items
Complete level 108 without pausing
Hit both targets simultaneously on level 28
Complete level 53 via the bottom-right corner
Hit 3 out of 4 targets on level 90 without using ladders
Get rid of all broken platforms on level 64
View the credits screen
Reach one target on level 70 without using gravity
Reach two targets on level 58 without using spades
Complete level 24 with the squares reaching their opposite targets
Pick up every extra item on level 82
Loot every vault on level 117
Complete all levels in mirror mode
Complete all levels in upside-down mode
Complete all levels in upside-down mirror mode