There are a lot of ways to hook up a light switch. That device assumes that the power feed is coming in from the top of the box and the wire to the light goes out through the bottom. If you don’t have that general arrangement then the device you have chosen won’t work.
It’s fairly common to run a wire from the light down to the switch, in which case you’d have the unswitched hot wire and the switched hot wire and nothing else (you wouldn’t have a second set of wires coming into the box). The electrician is supposed to wrap black electrical tape around the white wire to show that it’s not being used as a neutral in that case, but I’ve seen a lot of examples where the wires weren’t marked to code.
Another possibility is that there is a second switch somewhere that controls the light, in which case you wouldn’t have a simple SPST switch, but you’d have a SPDT switch instead. Again, the wires are supposed to be taped to mark that you don’t have a hot and a neutral.
Neither one of these seems to match up with the description in the OP though. But if you have either one of these then the device you bought won’t work.
You’ve got two hots (black wires) two neutrals (white wires) and two grounds (bare copper), with one set of hot, neutral, and ground coming through the top of the box and the other set coming through the bottom of the box. Is that correct?
When you measured “load” to neutral, were the neutrals tied together at that point or were you only measuring to one neutral? If so, which one?
Did you try measuring the neutral to ground voltage? If you get 120 volts there then they simply have the hot and neutral reversed somewhere on the circuit (not all that uncommon of a mistake). Check the voltage BEFORE you try to do a continuity check per Snnipe 70E’s suggestion. If there’s voltage present when you try to do a continuity check you can fry your meter.
If you have missing neutrals in your outlets you have a major problem. If you have missing grounds it’s still a problem, though not quite as bad, and either way it’s not to code.
If your location is correct, I’m somewhat in your area (I live in PA but I work in Owings Mills). I’d like to come out and take a look at your house some night after work, but I can’t do it any time in the next week or so. If you are interested, send me an e-mail (my address is in my profile). From your description, it sounds like you have a lot more going on than just a weird switch.