April Fool's day = April (Fool's day) or (April Fool)'s day
It's none of those, it's:
April Fools' Day
However, I understand it as a day for 'April Fools' rather than 'the day of Fools that occurs in April' because there is no Fools' day in any other month. And you can also call someone an April Fool.
If you must choose between them then take the latter, "(April Fool)'s day". But in fact there is always more than one April Fool, and 1st April is the day for all of them so
April Fools' day
which you can parse as "(April Fools)' day"