Eagle is… interesting.
First, to generate the Gcode, you will need a plugin, such as pcb-gcode. It isn’t terrible to use, but there is a bit of a learning curve.
To cut, you will need an appropriate bit. There are a number of options, but the most popular are vee-cutting bits. I use a 90 degree included angle, 1.5mm mill-drill bit with a 0.1mm point flat. Others use narrower angles, to make depth less critical. For the neopixel corners, you could likely get away with a 0.8mm (0.032") ball end mill or similar sized flat end mill. The traces are not that close. The larger tool will be useful for other things and less likely to break.
As for Eagle, you can copy sections of a layout and place as needed, but I have never found it easy. My solution would be to edit the gcode by hand (add a movement at the end, then duplicate the code that does the cutting) or use a g-code editor that has a step and repeat tool (I believe EditCNC does this, but have not used it). I believe some senders will do step and repeat it as well.
It might be easiest to start from scratch in a general purpose engraving package (harder)
Edit: reword for clarity and remove ref to carbide copper as it does not have a step-and-repeat tool that I can find. I was thinking of a different piece of software, but don’t recall what at this time.