Hello and thank you for contacting us. This is Dwayne B. and I’m an expert here and looking forward to assisting you today. If at any point any of my answers aren’t clear please don’t hesitate to ask for clarification. Also, I can only answer the questions you specifically ask and based on the facts that you give so please be sure that you ask the questions you want to ask and provide all necessary facts.
Certainly you can sue them but anyone can sue anyone for anything.
The key question is "can you win"?
To win a case like this you are going to have to have some kind of expert that will testify the work Firestone did on the power steering pump (or something else they did) cause the fire.
You can't sue for damages due to what "almost happened" but you can sue if you still have mental issues due to being in a car that was on fire. That is essentially what post traumatic stress disorder is about.
In a case like yours what you want to make sure happens is that the expert has a chance to inspect the vehicle, take notes and pictures, etc. before the insurance company repairs it, destroys it, or sells it and it is lost.
Please ask any follow up questions in this thread.