About the negation

There is the phrase : 1Nothing's gonna change my love for you.
=2 Nothing is going to change this = 3Nothing will change this. 1*,2,3=good English. *I am not sure of the status of 'gonna', and whether an examiner would pass it.
May I say that everything will not change my love for you? Yes.

Does ""no thing will change x"" = ""every thing will not change x"", logically? It is really a question of logic. (Imagine 10 different things acting on x, then imagine 0 things acting on x).

What's the correct form and why?

About the negation

There is the phrase : Nothing's gonna change my love for you.

'gonna' is a slang pronunciation of going to. You would not write gonna even though it might be pronounced that way sometimes.

May I say that everything doesn't change my love for you?

What's the correct form and why?

Nothing means none of the things. It means your love for the person can't/won't change. 

Everything means all of the things. It means your love for the person can/will change.