Within the Layer-3 MPLS Cloud itself (PE and P), typically it is either OSPF or IS-IS as link-state protocol that maintain some kind of LSDB (Link State Database) to map the entire network to support TE (Traffic Engineering). I never see the cloud based on EIGRP or RIP that have no concept of LSDB, not to mention network stability issue. Furthermore, EIGRP won't work if there are non-Cisco routers (i.e. Juniper) in the mix and RIP isn't scalable for large (cloud) network.
For CE and PE connectivity, typically it is either BGP or RIP. BGP is a standard choice since MPLS is based on MP-BGP, so that it makes sense to just use BGP (without the need to introduce another routing protocol on top of BGP). Some small networks without redundancy run static routes everywhere including traffic across MPLS cloud to reach remote sites.
Some managed services support EIGRP with client's network since the managed service includes CE-PE management. The EIGRP then takes place between CE (managed service device) and C (client's device). In this case, usually CE runs BGP with PE and simply redistribute the client's networks received via EIGRP into BGP.
It is also common practice with managed service to provide virtual default gateway via HSRP, VRRP, or GLBP to the client. Such virtual gateway provision could take place between redundant CE or between CE and C depending on agreement. Client has a choice to have all hosts as default gateway or to have C router with static route pointing to the virtual gateway.