Bill,
I think you may be looking at what are called captive panel screws. You can swage or put an E-or C-clip under the head to keep the screw from falling out, and retain the multi-turn fastener aspect. I cannot link directly with the McMaster-Carr item anymore, but here is the part number for an example 95966A121 at mcmaster.com McMaster doesn't appear to have countersunk varieties, but that and many other head styles do exist. They are a bit easier to install than the quarter-turn fasteners.
John