Should the noun following 'more than one' be in its singular or plural form?
e.g.
more than one day or days?

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Chop it up this way:

"More than / one _____"  So it's clear you use the singular.

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

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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.

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

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