While it is possible for a phone to not be capable of running on another carrier's/service provider's network, that itself seems kind of fishy. Sprint and Verizon are both CDMA carriers. So one's phones should work on the other's network. It sounds to me a lot like the person was trying to get out of giving you what you need, to try and talk you into having something else. Through them. Not shocking. They are sales people in the stores.
You will need a SIM through/from Sprint, but there is no reason you would need a newer phone. You may need to talk to someone else at the store (if you go back) even perhaps a manager, or just get the SIM and try setting it up yourself with them over the phone.
I really appreciate your response and many thank you's! I called Sprint over the weekend and they told me to get phone first then log in to activate. So I did. I was told I needed to go to the store and get Sim card. I could tell after waiting for 30 minutes the salespeople were all about iPhones. Finally after sitting down with sales rep he instantly looked at phone and said its a S5 and won't work but he will try. Scanned the phone and said nope, not compatible has to be S6 or higher since it was Verizon phone. He told me to ask my boss to give me an S7 or new S8 lol. Why spend 10 minutes for $50 helping someone get SIM card or spend 30 minutes or more getting iPhone and accessories. Seriously it's mind boggling watching people buy or upgrade phone.