Don't ever try opening an EPS file from Illustrator to Corel, or the other way around.
Just don't even bother with it.
RASTERIZE the artwork either in Illustrator directly, or you can attempt to copy/paste to photoshop or rasterize while opening in photoshop.
There is nothing difficult about this artwork as a raster image.
When trying to dig into the complex puzzle of a vector file supplied from a 3rd party, you're always going to spend a pointless amount of time to get through all the objects, settings, colors, etc.
Rasterize with a transparent background, and you've got everything as one image, 300 DPI at the final print dimensions so you don't lose any quality or detail that would have resolved on a stencil anyway, and you can then separate it in whatever program you want... photoshop, illustrator, corel, etc.
I see about 7 colors with some blending.
Yellow
Orange (or blend the red and yellow to get this)
Red (with some black mixed for shades)
Purple (with some halftone tinting)
Dark Brown-Grey (with some black mixed in darker areas)
Black
The dark brown is really going to be a difficult blend to create if trying to use the yellow/red/blacks to simulate it, but could be done with high-quality and mathematically accurate separation, film, and screenmaking/printing (this requires quality-control)...
The purple cannot be made by simulating unless using a Red/magenta and blue, which there is no other blue in the image so it's pointless to use a blue screen when you only need the purple for that blend.
The entire image could be printed as just Cyan, Magenta, Yellow - CMY process, but this is nearly impossible to truly get the blacks and shades to all behave correctly, and again you don't really need magenta or cyan... so this art is a mixture of spot-color modes and some blending / overprinting.
The dark brown is so dark that most of the black would have a tendency to overpower the lighter halftone of red/yellow that would create the brown when mixed with black, so this is why I suggest using an actual custom dark grey-brown ink and separating that piece on it's own.
You'll spend more time trying to do all this with vector, than with raster, and even more so when trying to translate it from Illustrator to Corel, as you've seen.
Vector is great if you're the one building your own artwork and you know how to use the color swatches and effects to create pre-separated files.... but it's a nightmare when dealing with supplied artwork, and I always rasterize and separate in a few minutes instead of waste time even looking at the vector objects.
Too many issues can arise especially ones you don't notice until the screens are made or it is up on the press, but all this is non-existent with rasterizing the original artwork.