Here is a piece that I wrote a while back explaining the difference between R. Elazar Ben Porta and R. Chanina Ben Tradyon.
On another note, Rav Moshe (Igros Moshe C.M. 2:74:2) uses the story of R. Chanina Ben Tradyon's execution to learn practical halachos. The complication with the story is that the Talmidim suggest that he open his mouth to expedite his death, to which he responds that it is forbidden. Yet, he himself makes a deal with the executioner to remove the soaked wool in order to expedite his death. How can he instruct the executioner to do that? Furthermore, Rav Moshe points out that it should have been possible for him to remove the soaked wool himself, why didn't he? It must be that it was prohibited for him to remove them since it would expedite his death. According to this approach, the question is even stronger, if he couldn't do it himself, how can he instruct the executioner to do it and even raise the fire? Rav Moshe explains that the talmidim thought it is permitted for him to open his mouth since if it were already open he could maintain the status quo and keep it open, to which R. Chanina responds that he can't actively open his mouth since right now it is closed. The removal of the wool would obviously be prohibited, and even the talmidim understood that. Rav Moshe explains that he wasn't concerned with the fact that a goy would be in violation of "retzicha" - murder, since it was ultimately beneficial. R. Moshe suggests that perhaps goyim are permitted to kill when it is beneficial for the victim such as expediting the death to relieve pain, and therefore wouldn't be any kind of lifnei iver to encourage the executioner to do it. R. Moshe deduces this from the subtle difference in language of the pasuk. The prohibition on a Jew to murder is לא תרצח which implies under any circumstances, but the prohibition on the goy is שופך דם האדם - spilling blood which is to the exclusion of euthanasia for his own good. Practically, Rav Moshe deduces from here that no action can be taken to expedite death even in the case of excruciating pain, but it should follow that to allow a non-jewish doctor to do the action should be permissible (but R. Moshe doesn't clearly articulate that practically it would be permitted by a goy).