Can I force customer registration at checkout?

Yes, you can force customer to register at checkout. If enabled, customer will not be able to checkout without having an account with your site. 

To enable, navigate to SYSTEM > CONFIGURATION and under Sales tab look for OneStepCheckout. Expand Registration section and change Registration mode to Require Registration/Login. Hit save button and refresh the cache management. 

SYSTEM > CACHE MANAGEMENT. Select all. Submit.