Collecting all Heart Squids but not trading them in means you will have to beat the game with only 2 hearts. This can make especially boss fights and later levels a bit though.
But remember one trick if you need fresh energy: Save your game, quit to the menu and reload the save file. You will have full energy again.
Also, while you are doing this you will be more or less forced to beat the bosses with no damage anyway. So you might as well aim for those trophies if you don't already obtained them.
But Just in case. Always have at least one of these items in your inventory for boss fights: Bubble Shield, (Super) Monster Milk, (Super) Pike Ball as well as food and an Auto Potion. Just in case.