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Since no other devices 'shut Off' the Verizon box's signal, I would say the problem is in the new Surface Pro 3 rather than elsewhere.
Check as to what channel the WiFi on the Surface is set to... it should be set to something fairly normal like 6 or 11. If it isn't try changing the channel to 6 or 11.
After that... you may simply have a defective Surface Pro 3, I'm sorry to say. If it isn't set to an eccentric WiFi channel choice, how it could be 'shutting Off' the wireless from the Verizon box is hard to guess, but doubtless is a defect, meriting its being returned for exchange.
(Please let me know how you know the wireless from the Verizon box has been shut Off... is it simply that other devices can no longer use the wireless from the Verizon box? or is it something else?)