Eigrp and tunnels are very stable. This issue of learning the end points via the tunnel can happen with any routing protocol. When this happens they flap constantly not just a few times a day.
So first is before the tunnel is built how do the 2 routers know to get to the end points.
The key here is that what ever method you use to make say ping work you need to make sure you do not damage that. Most the time this is done via static or connected.
If you learn the end points via EIRGP then you must be very careful to filter this network so it is not learned from the neighbor relationship formed over the tunnel.
Now if you have a /24 that contains the tunnel end point this whole subnet must not be allowed to pass the tunnel. You cannot cause some traffic for this subnet to be though the tunnel and the rest direct. The best solution is to use the loopbacks as the tunnel end points so you only have a single device to worry about.
The main reason to use a tunnel at layer 3 is because the intervening routers do not have routes to the networks on the end points. If you have full readability via eigrp already then you do not need tunnels but I would assume there is some requirement that cannot be met with eigrp or they would not have been created in your network.
