Yes, you can use other than a router to make cuts. We (and our DuPont certified fabricators) use jigsaws to cut Corian and other similar solid surfaces. Use the right blade. We also rip solid surface with table saws.
Use the router as described, with a straight edge, to get the straightest and cleanest cutout with an inside radius at the inside corners. You should never have hard, crisp, chiseled, right angle inside corners -- must be radius to avoid/ prevent cracking. You might also consider edging the opening with a heat reflective metallic tape on the cutouts edges.