You must be able to - I don't see Verizon listed among supported carriers for the "Apple sim" (or whatever they're calling it), and my current iPhone and iPad won't connect to the Verizon cell network without a Verizon sim.
In the past there were different hardware versions of Apple devices for Verizon and everyone else... I'd hate to see a trip back to that.
Likewise, and I think Apple would agree (better for their margins). I was quite pleased to see that all of the iPhone 6s, regardless of carriers, appear to be the same (at least as evidenced by there being only one version of iOS 8 to download compared to iOS 7 and the iPhone 5/5s)