There are actually two (possibly three) ways to go about this, depending on what you want to do...

The simplest is to put the Eero itself in "Bridge" mode. In this case, you'd still use your C1000A as your main router, which would hand out your IP addresses, and the Eero just becomes an access point system for your Wi-Fi devices. The downside to this is that you can't use features like Eero Secure, since it's not working as a router at all — the Eero becomes a "passive" system that really just connects your devices to Wi-Fi, and your C1000A router does all of the rest of the heavy lifting.

The second option is, as you suggest, to use "Double NAT," in which case the best thing to do really is to assign the Eero to the C1000A's DMZ. This will let you use it as a router, although you might run into some problems due to double NAT depending on what you're doing. These days this mostly affects VPN services, but can also still impact some VoIP and video calling apps. It won't impact normal everyday surfing and browsing at all.

To put the Eero into the DMZ, it needs to be on the same subnet as your C1000A's internal IP address (192.168.0.x in your example). You can either configure the Eero manually to use an address on this subnet (e.g. 192.168.0.2, as I'm guessing the C1000A is probably 192.168.0.1), or you can let the C1000A hand out the IP address to your Eero (via DHCP).

I'm not up to speed on the specific configuration steps for the C1000A, but it's probably easiest to configure the Eero manually to something like 192.168.0.2, since you'll probably need to assign the DMZ to a specific IP address, which could change if it's being assigned automatically. Once you have the Eero's "public" IP address established (e.g. 192.168.0.2), you need to log into the C1000A and find the DMZ setting and point it to that address.

This will have the effect of taking all inbound traffic that comes into the C1000A and sending it directly to the Eero, which at least means that it will be able to automatically open inbound ports for things like online games, VoIP services, and so forth using automated protocols like uPNP. If you don't do this, you'll likely have problems with these services.

The potential third option is to look for a feature called "Advanced DMZ" or some such. Again, I'm not a CenturyLink customer, but I've read several reports that this is a possibility on at least some of their routers, and it's what I'm using with my ISP (Bell Fibe in Canada). The "Advanced DMZ" feature lets the router actually assign the *public* IP address to the Eero, at which point it acts just like the C1000A isn't in the loop at all. 

Unfortunately, as I said, I'm not up on the C1000A, but if you do a bit of searching you may find some folks who have done this successfully... I know it's been discussed in the PPPoE feature request thread over in the "feature requests" section of these forums.