There isn’t a command to specifically inject a default route into EIGRP such as “default-originate”, instead the following methods are typically used:
1) Redistributing a default route, such as a static route.
R1#show run | i ip route
ip route 0.0.0.0 0.0.0.0 Null0
R1#show run | s eigrp
router eigrp ccie
!
address-family ipv4 unicast autonomous-system 100
!
topology base
redistribute static
exit-af-topology
network 0.0.0.0
exit-address-family
R1#show ip eigrp topology EIGRP-IPv4 VR(ccie) Topology Table for AS(100)/ID(1.1.1.1) Codes: P - Passive, A - Active, U - Update, Q - Query, R - Reply, r - reply Status, s - sia Status P 1.2.1.0/29, 1 successors, FD is 13107200 via Connected, FastEthernet0/0 P 0.0.0.0/0, 1 successors, FD is 131072 via Rstatic (131072/0) P 2.2.2.2/32, 1 successors, FD is 13189120 via 1.2.1.2 (13189120/163840), FastEthernet0/0 P 1.1.1.1/32, 1 successors, FD is 163840 via Connected, Loopback0
2) Using summarisation
The following network is a simple EIGRP domain. By summarising to a default network on R1’s interface fa0/0, R2 will receive a default route.

R1#show run | b router eigrp
router eigrp ccie
!
address-family ipv4 unicast autonomous-system 100
!
af-interface FastEthernet0/0
summary-address 0.0.0.0 0.0.0.0
exit-af-interface
!
topology base
exit-af-topology
network 0.0.0.0
exit-address-family
R2#show ip route eigrp
<snip>
Gateway of last resort is 1.2.1.1 to network 0.0.0.0
D* 0.0.0.0/0 [90/103040] via 1.2.1.1, 00:00:24, FastEthernet0/0
3) Static default to an interface (not recommended)
It is also possible to inject a default route into EIGRP using the network 0.0.0.0 command, if a static default route to an interface is configured. Note in the experience of the author, this feature is only supported in classic mode, and not supported in named mode. The Administrative Distance of a static route pointing to an interface is zero, and consequently EIGRP considers this to be a connected interface. As a result the network 0.0.0.0 command includes it.
The following network has a default route configured, pointing to fa0/0 on R1. This is then propagated to R2.

R1#show run | i ip route ip route 0.0.0.0 0.0.0.0 FastEthernet0/0 R1#show run | s eigrp router eigrp 100 network 0.0.0.0
R1#show ip eigrp topology EIGRP-IPv4 Topology Table for AS(100)/ID(1.1.1.1) Codes: P - Passive, A - Active, U - Update, Q - Query, R - Reply, r - reply Status, s - sia Status P 1.2.1.0/29, 1 successors, FD is 28160 via Connected, FastEthernet0/0 P 0.0.0.0/0, 1 successors, FD is 28160 via Rstatic (28160/0) P 2.2.2.2/32, 1 successors, FD is 28160 via 1.2.1.2 (28160/256), FastEthernet0/0 P 1.1.1.1/32, 1 successors, FD is 128256 via Connected, Loopback0
R2#show ip route eigrp
<snip>
Gateway of last resort is 1.2.1.1 to network 0.0.0.0
D* 0.0.0.0/0 [90/153600] via 1.2.1.1, 00:03:18, FastEthernet0/0
1.0.0.0/8 is variably subnetted, 3 subnets, 2 masks
D 1.1.1.1/32 [90/2662400] via 1.2.1.1, 00:03:18, FastEthernet0/0
Whilst it is possible to use this method, it is not recommended for two reasons:
Firstly, it is seldom a good idea to use an interface as the next hop for a default route, especially in a broadcast network such as Ethernet. An Ethernet segment may be attached to hundreds of devices all of which may be potential next hops. The router will perform an ARP for every destination it doesn’t know about to determine what the next hop should be. If this were a connection to the Internet, for example, their could be an inordinate number of ARP requests, which could impact CPU performance.
The following network illustrates this problem:

R1 has a default route configured to point to its outbound interface.
R2, R3, and R4 are receiving the IP 5.5.5.5/32 from R5 via EIGRP, however 5.5.5.5/32 is then filtered so R1 doesn’t learn it. Consequently when R1 attempts to ping 5.5.5.5 it has to ARP for it, and receives replies from R2, R3, and R4.
R1#show run | i ip route ip route 0.0.0.0 0.0.0.0 FastEthernet0/0
As the 0.0.0.0 network is configured (using classic mode) this is entered into EIGRP as a default route.
router eigrp 100 network 0.0.0.0 R1#show ip eigrp topology EIGRP-IPv4 Topology Table for AS(100)/ID(10.1.1.1) Codes: P - Passive, A - Active, U - Update, Q - Query, R - Reply, r - reply Status, s - sia Status P 10.2.2.0/24, 3 successors, FD is 30720 via 10.1.1.2 (30720/28160), FastEthernet0/0 via 10.1.1.3 (30720/28160), FastEthernet0/0 via 10.1.1.4 (30720/28160), FastEthernet0/0 P 0.0.0.0/0, 1 successors, FD is 28160 via Rstatic (28160/0) P 10.1.1.0/24, 1 successors, FD is 28160 via Connected, FastEthernet0/0
Pinging for 5.5.5.5 shows the first packet has to be ARP’d for:
R1#ping 5.5.5.5 Type escape sequence to abort. Sending 5, 100-byte ICMP Echos to 5.5.5.5, timeout is 2 seconds: .!!!!
The following packet capture shows the ARP being processed by all three routers:

The second reason, is it is generally a bad idea to use the network 0.0.0.0 command in itself, as EIGRP will try to establish and adjacency out of any interface, potentially leading to unanticipated neighbour adjacencies.
References: