IMHO - If he was working on company time for the benefit of the company, then the company owns the information. I see it no differently then when you design something for a company while in their employ - that is your job. If it were otherwise, there would be a lot of wealthy scientist at [former] Bell Labs, 3M, IBM and such.
Surely Twitter owns the stream, when it comes down to it? Obviously it has value, but it's out of Phonedog's hands because it's on an external website, and Twitter can do what they like.
This is what can happen when you invest significant resources in developing something you have no direct control over. If only there were a way to keep your company announcements and customer service chat on your own web property...
It must depend on the terms of his contract of employment. Most employment contracts have probably not kept up with social media, but may have something general about IP developed in an employer's time... Interesting argument... lots of fees for lawyers!