Instead of punching a hole, use an Xacto type knife to cut an x about the size of the eyelet. Put the eyelet through the x by inserting it, of course, through the front of the whatever you are creating so the paper pushes to the back.
Turn your creation over. Find a Phillips head screwdriver that just the tip, but not the rest of the screwdriver part, fits in the eyelet hole. Use a regular hammer, or a lightweight hammer (you can smaller light hammers for .99/$1 at dollar places, to hit top of screwdriver. If only part of the eyelet set, take screwdriver out and reinsert it so the outer edges of the screwdriver bit goes against the sides that have not set yet. Hit it. Take screwdriver out and hit the eyelet itself a few times....works like a charm!
Here is a hint about the screwdrivers: Get a six bit screwdriver (one that you take the bit off and change it) so you have a variety of sizes right there). Most regular eyelets can be set with this. For tiny eyelets, get a precision screwdriver set (the tiny ones like for glasses)and use the phillips head ones out of it.
Oh yeah, if you use a glass cutting mat to cut the x with, don't forget to take it out! LOL..had that happen to me cause phone rang....crack!