Howie, although you certainly can install laminate overtop of the existing vinyl, I would recommend that you remove the vinyl and the
1/4" plywood underlayment before you install the laminate. If you don't, you will have a nasty transition between the new laminate floor and the existing floors in other rooms. You'll also have to re-fit around door mouldings, doors, etc.
Remove the screws you put in, they are too short to do any good--they are only 'biting' 1/2" into the joist, if they are even hitting a joist. Removing the underlayment is easy. Score it down the middle of the room with a circular saw set to about 3/8" depth, and use a cheap blade cause you may hit the odd nail. Then just pry it up with a crowbar...it's so thin it's a piece of cake.
Once you have the underlay removed you can see where the subfloor nails are for sure. Run in a screw about every 4 to 6 inches along these nail lines. Use 2 1/2 inch screws and make sure they are countersunk below the surface of the subfloor.
You can put your laminate flooring right on the subfloor. Make sure the subfloor is solid (not bouncy). If there are big cracks, use floor leveler and sand it smooth. You won't need to bother filling small cracks or screw heads, the laminate is pretty forgiving.
I just finished this process with cork-faced laminate. The squeaks are all gone, and the floor is magnificent.
Good luck.
Mr Fixit eh