After much contemplation I came up with the following technique I think worked out pretty well.
After removing the ugly light I used a razor blade scraper (like for paint, or scraping bugs off the windshield). I carefully cut off the protrusion left by each screw when removed and the ridges left by the light corners.
When the ridges are cut off it leaves a little sliver of underlying ceiling board showing. But that will be covered up later.
Next I used a center punch and a couple of taps with a hammer to just slightly indent the screw hole a bit. This gives the filler something to fill in and ensures the area around the screw hole is at least flat or slightly indented.
I found this hole filler kit at the hardware. It has a tube of filler, a scraper and a sanding block all in one. (Wish I had this when I lived in a college dorm, better than toothpaste).
Regular Sheetrock filler and sandpaper would work just as well. I filled each hole, let it dry/set and gently sanded it flat. You can gently sand the vinyl ceiling without damaging it.
Last step was to touch up with some paint. I found two small bottles of acrylic paint at an art supply store. A "Mushroom color (slightly off white/tan)" and a White color. Mixing them together I was able to come very close to the existing color of the vinyl ceiling. Now the holes are almost invisible and you need to know they are there to see the patches.