After instructing his viewers to boycott a product a bombastic news anchor is held hostage by a fired employee who is determined to make him pay with his life.

Agreed with Foxtrot25 and Richiev.

You can start the logline with him being taken hostage but you need to describe what he must do to resolve the situation. I would add that you could give him an unusual ability that he could use to free himself. What if he is an ex-cop, a veteran, or former MMA fighter? Anything really as long as it gives him a skill he can use.

The first half of the logline is not needed. You should drop the first part then add what Foxtrot suggested.
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"Held hostage by a disgruntled ex-employee who's lost his insurance, a bombastic conservative news anchor is given 12 hours to convince the American people free healthcare is a right or be shot live,?on the air."
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This example would read better if you rewrite it focusing on what the victim/protagonist needs to overcome and a little bit more of what he needs to do for it. A ticking time bomb would help too.