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