Their website is kinda weird. The activation page you link has instructions for finding the IMEI on Android devices. But if you go through the "
shop" link on the same site their BYOD programme appears to be for iPhones only! But I think it's the second that is correct: they used to offer BYOD on Android, but it looks like they currently don't: you can buy an android phone from them, but I don't think you can bring your own. Not absolutely certain, but that's the impression I have from what I've been able to find.
As for why it might be incompatible (apart from just their only accepting certain iPhone models), exactly what model of phone is it? Not just "S3" or "S4" but the full model number. If it's a GSM model then it's not compatible with 2G/3G, since Virgin in the USA use the Sprint (CDMA) network and both S3 and S4 (I don't know which you have, you mention both) had separate CDMA and GSM models. And for 4G as far as I can tell no S4 was compatible with the 4G bands that Virgin use (I don't believe the S3 had 4G at all). So depending on which model you have it might have limited compatibility or none at all.
As for tricking their system, that's not likely to work. If it was compatible with their 4G then maybe putting a SIM in is all that you'd need for that (though they can check IMEIs and bar unregistered ones if they choose, so no guarantee). But CDMA networks don't use the SIM for 2G/3G, so I think you'd need to register the actual phone ID for that, which means that no other number will work (like most of the world, we don't have CDMA networks where I am, so I may be out of date on how LTE-capable CDMA phones interact with the SIM for the CDMA part, but I don't think you have one of those so don't think that question arises).
So I doubt the phone is compatible, and if it is compatible to some extent I doubt you can trick them into registering it.
You might be able to use it on a different service, of course.
