When you go back to re-meander (is this a word??), start a half inch or so back and go over the stitching. Then when you get back to the stitching again, go a bit further over the stitching. That should secure the stitching.
Also, does your machine have a little lever for increasing or decreasing the speed? Be sure it is on the fastest setting. You might try experimenting with increasing the top tension and see it that helps, but eyelashes tend to happen on curves because you will have a tendency to speed up as you make the curve. Keep the machine at the same speed and move your fabric at an even speed. Some people listen to music to keep their speed even!