Yes, the floor should be thoroughly screwed on every joist. In addition, construction adhesive should be applied to every joist before the subfloor is laid down.
As far as the cost increase for the upgrade to plywood, consider that OSB doesn't have the T&G profile of subfloor grade plywood. In addition, like Jay said, you didn't mention how many square feet of floor we are talking about here, but you should be able to figure out if the cost is inappropriate by pricing the difference in their costs at home depot. Make sure you compare apples to apples, and allow the contractor the ability to make a little bit of money in the process.