Legion's Initiative vs Day of Judgment
You sure can. 

The stack will also matter for this because otherwise your creatures will go poof. Basically in response to them casting DoJ, you pop your Legion's Initiative to exile your guys. There's no timing restriction on Legion's Initiative so as long as you can pay for it, you can do it.

Or in listed steps:


Day of Judgement is cast and opponent passes priority (most likely onto an empty stack because it's a sorcery but there's ways to make it instant speed but that's irrelevant here.)
You get Priority and pop your LI, putting the ability on the stack
Priority Wave
LI Ability resolves, all your dudes go away until the next combat
Priority Wave
DoJ resolves, blowing up the remaining dudes on the battlefield.