I make a lot of jewelry and fused glass, too. While it is important to hold the object you are drilling securely, another thing that is happening is that friction, (drill bit going through metal or glass) is making the object hot and hard to hold on to. For this reason, you want to hold the object in a small dish of water, or dip object in water every 10 seconds, or use a lubricating product, or even beeswax on the drill bit.
I have found 1 or 1.25 mm diamond drill bits (they cost about $9 for 5) are the best for making small holes for jump rings to fit through. I use a product called "Burr life" as a lubricant. Both of these items I order from Rio Grande. But maybe you are going for larger holes?
Also, please wear eye protection and tie back long hair. Better safe than sorry.