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