Randy - I have never used Rhino, but there might be some issues with it regarding it being a 3D program and exporting to 2D DXF. I know a few people who use it successfully when creating 2D artwork similar to what I do. I am pretty sure someone else using Rhino was having some DXF issues with Lightburn.
I created another “Houdstooth” pattern in Draftsight 2D CAD, and trimmed out the excess near the circles, and then created polylines of it all. I imported the DXF file into Lightburn, and set the DXF auto-close tolerance to 0. Lightburn imported it correctly. The repeating pattern looks a bit odd near the edges of the rings because a lot of pieces have to fall out in order for it to work.
I also imported the DXF file from Draftsight into an older version of a free program called nanoCAD 5.0. I exploded all of the polylines, and recreated new polylines from the lines, arcs, and circles using the Boundary command. nanoCAD is a typical AutoCAD knockoff (just like Draftsight) with it’s native file format either DWG or DXF. I generally use release 13 DXF. When I exported a release 13 DXF from nanoCAD and imported it into Lightburn, it lined up exactly to the DXF data I originally exported from Draftsight.
What you might want to do is import your Rhino DXF into nanoCAD, explode everything, and then use the Boundary command, and see if it creates linked polylines of the entire file. Just put your original data on a separate layer to delete later. Draw a rectangle around your imported DXF data, and then use the Boundary command. If you need any help PM me. I think you were pretty close in the image you referenced as being better. Lightburn does a good job of importing polylines, and if the CAD design is correct, you don’t need Lightburn to connect anything.