Here is the full list of all 15 Squares trophies - there are 13 bronze, 1 silver and 1 gold.
Finish a level in the last tenth of a second
Get 3 Squariness in a level
Finish all levels with 3 Squariness
Finish the game
Beat the first 45 levels
Tap 500 blue squares
Tap 1000 blue squares
Tap 2000 blue squares
Swipe 250 squares
Swipe 500 squares
Tap 100 red squares
Lose a single level 25 times in a row
Finish a level in under 1 second
Create a level
Play a level you created