'I am going nowhere' or 'I am not going anywhere'

Solution 1:

The sentences are perfectly correct and mean technically the same thing but carry different overtones.

I'm not going anywhere is a simple statement of fact:-

Will you still be here when I get back?

Yes, I'm not going anywhere.

Whereas I'm going nowhere would be used when more emphasis is required; in an altercation in a shop, one would say

I'm going nowhere until I get my money back

or when in a despondent mood:-

My life is a wilderness of despair. I'm going nowhere.