CasperJS is a higher-level wrapper for PhantomJS, so - yes, it could be done with PhantomJS solely... But you wouldn't want to, because CasperJS makes automation easier.

Are there any libraries to facilitate database-connectivity to SQL Server or MySQL from javascript? I've used CasperJS to scrape some sites, but always fall back on post-processing the scraped data with another program in order to get it into my database. I'd love to be able to do it all from one piece of code.

You could always have your CasperJS scraping script make an AJAX request to a RESTful API for your MySQL DB. You won't be doing it all from one piece of code, but you'll be doing about 90% of it.