If you are using twitter as a broadcast to the World that you exist and are available for flirtatious exchange- then of course it's going to cause anxiety to your partner.
BPD is a fear of abandonment. Do you see how your need to flirt (and I assume it's a need) might be a trigger for his fear? I do.
In fact, if I knew my partner was Borderline on this, I would try to help them understand that their feelings of dread and fear are unfounded. But that's if I was staying with them and making a conscious effort to work through the insecurities. In fact, I probably wouldn't be on twitter in the first place if it caused such a fear- unless I was working with my partner in therapy. When it comes to twitter or love, twitter and the need to tweet- would be second place.
Well, and obviously you didn't know about this aspect of it, was that he would be on a site (for bears) talking to guys to get his rocks off. He liked to go on there, and not just flirt (my biggest comment to someone was, "WOW yer really cute!" but he liked to actively engage in conversations about what he'd do when he were with them, etc. I am a computer guy (network admin) and at one point, I suspected he was up to something and decided to remotely watch what he was doing ON MY COMPUTER mind you. He was on that same site, while hiding it from me, (including using the "incognito" browser portion of Chrome) asking a couple guys if they were ever free during the daytime to hook up, etc. Mind you he wasn't' working at the time, I was supporting him, and he was asking guys if they wanted to hook up. In my apartment.
With all respect to you and your situation, I think anything that I did to him (a little flirting openly so he could see it while still supporting him and making sure he knew I was with him and only him. The guy I was flirting with lived 2000 miles away.) is far more harmless, regardless of mental disorder, than asking guys if they'd like to hook up during the day while I'm at work. And doing said chats behind my back - whereas mine were very much in the open. Until close to when we broke up and I closed my twitter and blocked him so he couldn't see what I was doing anymore. And we were broke up.
And regardless of our status back then, we've been broken up for 3 months. He created this fake account a month ago and has been trying to pump me for information. Sorry if that wasn't clear as well.