In my understanding of the term, civic republicanism refers to the idea that we are supposed to have leaders who will work together towards the common good. It is the belief that this sort of a situation is possible in a society like ours.
Civic republicanism, we can argue, is something very different than what we have today. It seems that what we have today is a system where leaders in both parties try to seek their own political ends without much care for what is best for the country as a whole. This might give you one way to teach about this idea. You might, for example, give the students some basic facts about the Social Security system, highlighting how it will cease to be able to pay out full benefits in the relatively near future. You could give them options for what to do about it, like raising retirement ages, allowing more of the richest people's incomes (that part above the current cap) to be taxed for Social Security), reducing benefits, things like that. Then ask them to come up with compromise plans. Then you could compare that to what is happening today with the intransigence of all sides.
This would seem to be a good way to teach about the need for civic republicanism.