There are a couple of aspects of your question that puzzle me. Especially where you state that you do not want any static route at your site or at your client site, but then ask if you can use static routes and EIGRP to accomplish your goal. Do you intend to use statics or to not use statics?
I am also a bit puzzled when you talk about an external client and then about using EIGRP. EIGRP is usually treated as an Interior routing protocol which means it is usually within your enterprise not to external entities. There is nothing to prevent you running EIGRP with that client but I suggest that you think carefully about how much of your internal network you want to advertise to this external client.
If you do want to advertise the NAT address to the external client there are a couple of ways that you can do this. Probably the most easy way is to configure a static route for the NAT address and to redistribute static into EIGRP. If you have several static routes you might want to consider using a distribute list to control which of the static routes get redistributed. If the NAT address is part of a subnet within your network it may be possible to have EIGRP advertise that subnet to the external client. It is not quite the same thing but some times when I have a pool of addresses that I want to advertise I will define a subnet on the loopback interface (even though the loopback is really a host address) and put a network statement into EIGRP to include the loopback. This gets EIGRP to advertise the pool of addresses. You could probably do something similar in your situation.
We are pleased to announce availability of Beta software for 16.6.3. 16.6.3 will be the second rebuild on the 16.6 release train targeted towards Catalyst 9500/9400/9300/3850/3650 switching platforms. We are looking for early feedback from custome...