The problem is, as you accelerate past the speed of light, at the moment you are traveling at the speed of light, you will have traveled an infinite distance in an infinitely short period of time. Right? Therefore, you would have to be externally accelerated.
Right, in the sense that this is one way in which you can think of why this scenario is not possible. When you reach the speed of light, you are traveling an infinite distance in zero proper time -- meaning that from your point of view, you travel infinitely far instantly, and to an external observer (not going at the speed of light), your clock is no longer ticking. How can you accelerate, if in your frame of reference you do not experience time? What does acceleration even mean if you experience no time?
Actually, it's even worse than that, because being accelerated externally does not work either. How can you accelerate something externally if all possible signals and interactions which could cause further acceleration are moving at the same speed or slower than the thing you're trying to accelerate?