Well, sort of.  This is correct:

    One day in high school I remember thinking that if I didn't go into show business I was going to become a fireman.

This is putting this sentence into the past:

    I think that if I don't go into show business I'm going to become a fireman.

So, to make it correct there is a lot of implied context, meaning that it's informal but not incorrect.

Your sentence is awkward.  I don't remember seeing "I would have been going to..." before more than maybe a couple of times in my life.  I would say "If I hadn't gone into show business I would have become a fireman."