Chop it up this way: "More than / one _____" So it's clear you use the singular.

Definitely one day. One days does not make sense in any case I guess.

More than one day. (this is the right one) "More than" suggest plural. " days" is also a plural. So we don't double the plural effect.

Leigh is right - in your example it should be - more than one day.

It should be in it's singular form buddy. For example I bought more than one apple (not more than one apples).