I have a layer 3 3750 advertising its vlan interfaces via EIGRP to two routers. R1 and R2 connect to MPLS via 2 different carriers using BGP. R1 is the preferred path. My question is, how do I configure a default route for Core1 using EIGRP that will be aware of failures on either of these carriers.
Perhaps you could investigate using ip sla for your needs. These links may not be exactly what you are searching for but it could help with searching on ip sla.
You can't achieve the function you want with just a default route on Core1.
You might inject BGP routes within EIGRP on routers R1 & R2, and define a different weight to these 2 EIGRP neighbours at the Core1 level.
If either R1 or R2 fails or their BGP neighbour fails, then all the traffic will use the remaining one once EIGRP will have updated its routes table.