I don’t really know much about the world of backlinks and SEO. Honestly, I don’t believe in direct attempts to influence such things. Whether posting links directly in your tweets, using a specific application to make those tweets, or otherwise has any influence on these things or not is way outside of my knowledge domain. I would consider any resultant SEO bump as side effects of producing engaging content that is further shared by others.
The types of questions I would ask are: When you tweet from your account, would a link back to a specific page on your site add value to the content (the tweet)? From your site, do you provide a means for users to tweet about the page they’re on (Tweet Button) or to follow your account to stay abreast of tweets they would find relevant?