Some of the same arguments that apply in the abortion debate apply here, because you have to start out by demonstrating the value and dignity of life. In my opinion, if someone doesn’t believe in God that becomes too subjective to be logical, but there are significant philosophers who disagree.
Once you have demonstrated the surpassing value of life, then both kinds of euthanasia become violations of the dignity of life. Involuntary euthanasia becomes a form of murder and violates the principle of respect for a person’s free will, while voluntary euthanasia, or assisted suicide, becomes an abandonment of one’s duties. Every person has a responsibility to do whatever good we are capable of; suicide abandons this responsibility, and is therefore bad.
In the U.K. the Church has been fighting an effort to legalize assisted suicide, and the pro-suicide argument is that this is a matter of free will and conscience. If you feel that your life is worthless, they say, the dignity of life argument no longer works, and to enforce that is a violation of your conscience. One of the Church’s responses to that is that it’s not just your conscience that is involved here, because in assisted suicide you bring a doctor in as well, along with the state’s medical standards. And even if you think that your life is worthless, the State’s standards have an obligation to be based on the truth, which is that all life has enormous value. They therefore have to hold their doctors to that standard, because everyone has to pay attention to the truth and hold firm to that, not just do whatever you want them to do. I think this argument is effective because it grounds the objection to assisted suicide in an objective responsibility to the truth. So keep that in mind as well.