AppleCare is not on Twitter as far as we can tell. Either that or one of our researchers determined that they don't really help customers through their Twitter handle. Either way, you're better off looking for a different way to get in touch with them. If it makes you feel better, a lot of companies that do have Twitter handles still redirect customers to another form of communication in order to help them with customer service issues. In this particular case, our advice is this: Call AppleCare.
While we don't know of a Twitter username for AppleCare, in general we do support helping you find a way to contact companies over Twitter for the purposes of handling customer service problems. With our problem-specific help tools, we help you create a page for your issue and then one-click to Tweet the company a link to it if they have an available Twitter or DM handle. The nice thing about it is that you can share that page with friends and other customers to show the company that others are paying attention as well, which we find gets you a faster, better response in most cases.
Of the 5 way(s) to get in touch with AppleCare, customers like you claim that the best one is the 800-275-2273 option. From this page you can easily access all the possible contact information options for AppleCare as well as read up on common problems that other customers are having, find out how to fix the most popular ones, read reviews, and more. Know of any other ways to reach AppleCare? Pretty please let GetHuman know about it. We work hard to bring you as much free information and tooling as we can, and we grow that toolset every day, bringing things like the call-back tool that skips the hold music when calling a company or call-recording for customers (beta). The more you use GetHuman and tell friends about it, the more we're able to provide useful, free tools for you and customers worldwide.