switching to sprint

I'm sure Apple would have no objection to using any carrier, but you really need to speak to Sprint and Verizon

No, not easily. In order for a phone to be activated on Verizon's network, it generally needs to be one sold specifically for use on Verizon. If you have a SIM that is already active on Verizon's network and is the right size, you may be able to activate a non-Verizon iPhone. However, you will only be able to use the phone on the 4G network. If you go anywhere that only has 3G, the phone will not work. You may also have difficulty with some features. And, of course, all of this is assuming that the Sprint phone is unlocked.