I think in theory EMGs should actually give you less feedback than most other pickups since they have a preamp built in that's supposed to reduce things that cause it. I'd take a look at the wiring of the guitar in more detail, make sure all the connections are solid and the soldering is good. There shouldn't be much soldering for an EMG equipped guitar but there will be some. If you're not comfy with it then take it to a tech and have it looked at.

It's also worth making sure that everything else in your signal chain is all right, everything from pedals to cables to the amps you're using, everything. Make 100% sure it's definitely the guitar. It could be almost anything in your signal chain causing it.
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Doubt it's the pickups. Even busted wiring will usually just mean a loss of signal - not crazy feedback. More likely to be something between the guitar and amp, if not the amp settings. A bit more information is needed if you want help figuring this out. Does this happen when you plug directly into a clean amp on low volume?
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Yeah, start with plugging directly to the amp.
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