We had a situation the othe rnight where fire spread onto the hex occupied by one of my units engaged with one of the enemy. My opponent argued that he shoud become "unengaged" now that my unit was shrouded in flame but I couldn't find anything in the rules of FAQ to support that.

In the end we played it that they stayed engaged.

I can't say for sure but I've found that most rules questions that come up are usually solved by taking the game very literally and not overthinking it. If the fire rules don't mention becoming unengaged then you're probably not unengaged.