This is one of the few areas where I seem to disagree with Sprout, but that’s likely due to what we’re used to and our work/travel/lifestyle requirements.
A good laptop makes a great workstation when connected to a large desktop display. For typical graphic design work, a desktop machine has no particular advantages other than significant cost savings, which, depending on your budget, might be the deciding factor.
What a laptop does offer is portability and a second display when connected to a desktop monitor. If these two things aren’t important to you, a desktop machine is probably the best bargain since they cost less. All things considered, I’d personally go the laptop route.
As others have said, 16GB of memory is probably adequate — for now. If you can afford it, though, I’d opt for 32 because you’ll likely need it in two or three years. As for disc space, a large SSD is great when paired with an external drive for storage (I don’t keep many things other than applications on my computer’s SSD). A fast processor is also important, but there’s no need to go overboard with it. The extra cost of the fastest processors doesn’t result in noticeable performance enhancement on typical graphic design projects.
You mentioned mostly using Illustrator and InDesign. If that’s the case, the specs I’ve mentioned above are enough. For that matter, they’re also enough for Photoshop. If you’ll be using Premiere extensively or regularly working with animation or 3d modeling, you’ll need more of everything.
I see no need for an eGPU for typical graphic design work — seriously, no need at all. If gaming is a big part of your routine or, as already mentioned, video editing and 3D modeling become mainstays of your computer use, it might be something to consider. Otherwise, I’d put the money saved into more memory, additional external storage/backup, software, or peripherals, like a good printer or drawing tablet. Even a more comfortable chair seems like a better investment than an eGPU.
I have no idea how you’re managing on a 1.10 GHz, 4 GB RAM machine. That’s barely enough for the operating system.