To what ends are we discussing choice in this matter? To show that suicide is somehow less of a problem because victims are "choosing" to take their own life, or is this purely a semantical argument? In either case, it's irrelevant.
Let's begin with the premise that no one under normal circumstances
wants to die. I find this to be self evident, so won't get into discussing it further.
That being the case, even if someone "chooses" to take their life, it follows that dying isn't their ultimate desire. A terminally ill cancer patient may over-dose on morphine because they can't bare the pain, but their desire is to end the pain, not end their life. Likewise, a single parent might sacrifice their life for the life of their child by jumping in front of oncoming traffic to push their child out of the way, but it was not the parent's desire to die and leave their child orphaned; they made that "choice" because it was better than the alternative. Similarly, a very depressed individual might take their life because it's the only way they see to end the pain; they don't want to die, but they see it as the only way to end their suffering, so they "choose" to die.
...
As an aside, I feel anorexia and bulimia are similar in that the sufferer is "making a choice" not to eat properly; it appears that way to us and is therefore frustrating and heartbreaking to watch, but it's not really a choice they are making consciously. From their perspective there is not other option, and is therefore not a choice at all.
*****
To the original question... mental illnesses are drastically complex, so I'm very hesitant to make a generalization like "all suicidal people are depressed." This wouldn't allow for other causes like anxiety disorders, schizophrenia, pain management, etc. It also wouldn't allow for the varying causes (and therefore ultimate diagnoses) of of depression; there are biological and chemical causes, social causes, seasonal causes, drug and alcohol abuse, etc.