Trademark is the normal legal foundation for protecting a name. But trademarks are limited by industry and geography, so attempting to enforce them on a single global namespace like Twitter or DNS is always going to end up with conflicts.
However, this is probably the best way to protect a name you use on Twitter: register a trademark and let them know that you're using your registered name as a Twitter handle and want it maintained for your business/organisation.
However, this is probably the best way to protect a name you use on Twitter: register a trademark and let them know that you're using your registered name as a Twitter handle and want it maintained for your business/organisation.