If the phone is under contract with a different company, then the phone is probably also locked. To get it unlocked you would have to contact the prior company (Verizon) and have them unlock it. However, if you owe them for the phone or other things, they may well not unlock it for you. They are not required to do it if you don't own the phone. Now if you own the phone (no payments due on the phone) but owe them for something else, then they should not be able to tell you they won't unlock it. I mean they can try, but in such a case I would speak to someone higher than the standard CSR (Customer Service Representative). They have to unlock the phone if it is fully paid for.
Now in regard to using the phone on a different carrier, once it is unlocked, Well that depends on if the different providers are compatible. Verizon is a CDMA carrier, as is Straight Talk. Straight Talk uses CDMA in some area and I believe GSM in others. So your area may not have the CDMA option. I would suggest calling them again, and speak to someone other than the general CSR that answers the call. They may not even know the information I am giving you here, as they are often just given limited info.
If, whoever you talk to, still insists that it can't be done, there are other companies out there that are CDMA.