From what I understand, FMLA protects you for 12 weeks of unpaid leave within a year of the child being born. And that protects your insurance. Not the paid leave. This is assuming that your company is required to comply with FMLA (50 employees within a certain radius?)
Most insurance is paid at the beginning of the month and runs through the end of the month. So if you wait until at least the beginning of December to give leave, even if they fire you immediately you'd still be covered under their insurance until the end of the month. And then there's the whole FMLA thing, so I don't imagine that they'd want to mess with that and contest it, but you know more about that than I do. And then there's COBRA.
I'd call/email in 2 weeks notice. Does your boss already know or are you just assuming he won't care because he's leaving?