Beautiful suit - and pretty kitten 
So yes, the 
does a fabulous job on EVA. You can do a search for it and find other instances, but this may come in handy: EVA Foam Settings for Cosplay Projects. Obviously you’ll want to run some tests on the material you’re actually using (after you’ve confirmed it’s laser safe and has no chlorine or vinyl in it). EVA tends to pull back ever so slightly from the cut - so if you want something that’s actually 1" square, you’ll cut your design to 1.025" square (you’ll know that specific number after you’ve run your tests). You’ll see discussions on here about kerf - which is the amount of material the laser evaporates - this is similar.
I generally find the easiest way to do it is to hand sketch the basic shape you want - then scan that into your computer - then use your graphics program (I use Inkscape) to draw a “perfect” line for the to cut. Things like the eyes and paws you can do one and then flip over for the other side - while things like claws you can use a single claw and cut it out 16 times.
Another bonus is once you have a piece that (for example) fits your hand - then if you want to do a completely different suit you can still start with that as the base since you know it fits!