Here is the full list of all 13 SquareMan achievements.
Fall into the Caverns for the first time.
Hang onto a Climbable wall.
Get hit by a Spike to do a 360 spin.
Attained Willpower to save yourself from Insanity.
Made it out of the Caverns.
Passed the jump off the ceiling into the GreenWoods.
Escaped The GreenWoods and entered The Dead Hang Valley
Dashed for the first time.
Surpassed the Dead Hang Valley and made it into the Dark Blue Castle.
Conquered the Dark Blue Castle and set foot on the Perilous Mountain.
Escaped The Path Of No Return with Willpower and crossed the bridge to freedom.
Kneeled down to the Statue of the Eagle. Failed many times or not, and escaped The Path Of No Return.
Beat the game without hitting a Spike or Slope