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."

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."