Hi there, interesting question.
In the negative form 'about to' still gives a very similar meaning, which is that something happened a very short time before you were going to do something else.
"I was about to open the door when my phone rang." = a very short time before I opened the door, my phone rang, so I didn't actually open the door when I wanted to.
"I was about to hang up the phone when I realised I still needed to ask what time we were meeting." = I realised I still needed to ask something, so I didn't hang up the phone when I wanted to.
So in the negative sentence, as Zowee mentioned, you are talking about something you didn't want to happen or you didn't want to do at that specific time. Usually after using 'I wasn't about to...', we add a 'so I ...' clause to show what action we took instead.
Here are some examples:
"I wasn't about to let him have my children, so I held them behind me to keep them safe."
"I knew that I was losing the fight but I wasn't about to give up, so I put all of my energy into the last round and managed to knock out my opponent."
"I felt so ill when I went into my maths exam, I couldn't stop sneezing. But I wasn't about to fail the most important test of the year, so I ignored my runny nose and tried my hardest."
Hope that helps :)