Daniel, you could redistribute BGP into EIGRP with a route-map permitting only the default route. On your EBGP peers the default route will be learnt via EBGP (as long as next hop reachability etc) admin distance 20 (better than eigrp external 170) but you should still see the default route in the eigrp topology database in the event of a bgp peer failure.
On you non-BGP routers you should see the default route via eigrp external.
Example config of the BGP into EIGRP redistribution
router eigrp 1
redistribute bgp 65000 metric 10000 1000 255 1 1500 route-map default-only
network 20.0.0.0
auto-summary
no eigrp log-neighbor-changes
ip access-list standard default-only-list
permit 0.0.0.0
route-map default-only permit 10
match ip address default-only-list