Utilitarians believe that the most ethical action is the one which maximizes happiness.
Depressed people are often so sad and unhappy that one might say that their very existence offers the world a negative amount of happiness, such that the world would in general be more happy if they were dead.
Thus, do utilitarians believe that we should kill depressed people?
I'll assume you want to analyse this within a rule-utilitarian framework [*]. On that assumption, I think you can refute the proposition "rule utilitarianism implies depressed people should be killed" using an instance of a fairly general argument template, regardless of what utility measure you prefer. The general idea is to argue that even if there were positive gains in utility following from implementing the rule, the "indirect" suffering caused by living in a society where such a rule was implemented would greatly outweigh the "direct" positive gains (again, assuming such gains existed).
Assume for the sake of contradiction rule utilitarianism implied the rule "depressed people should be killed". Consider that the lifetime prevalence of mood disorders is about 20%. Hence, (on our assumption) reasonable persons should judge it relatively likely they could be killed because of depressive episodes. The angst and worry following from the prospect that one had a fair chance of being killed because of depression and that one with virtual certainty again and again would see friends and family be killed because of depression would presumably lead to great suffering in the general population – significantly outweighing any potential "gains" reaped by killing the depressives. (For even more support for the proposition that the net utility delta is negative, you could also consider the fact that most cases of depression are treatable.) Such a rule, which greatly promotes suffering, cannot be a rule of rule utilitarianism.
[*] I'm pretty certain you can give a somewhat similar argument in most act-utilitarian settings, but that's a different story.
There is no canonical formulation of utilitarianism. Classical utilitarianism - that of Jeremy Bentham and JS Mill - ran on happiness. Happiness was the moral metric, so I don't see why we shouldn't address your question as it's framed. (I agree with Conifold that a formulation in terms of utility is preferable.)
If unhappy, depressed persons were killed, we do not know how much further unhappiness would be caused by the killing. The resulting unhappiness of family, friends, neighbours and others connected with the individual might create greater unhappiness than would the continued unhappy life of the individual.
A depressed person is unhappy at a time but not necessarily at future times if allowed to live. Her total future happiness might be greater than the unhappiness they presently endure.
You are committing yourself, so I read your question, to a rule : 'whenever happiness can be maximised by killing the depressed, then kill them'. (Check the difference between rule- and act utilitarian, widely explained online.) We have to consider the consequences for happiness of this rule. It would maximise happiness only if the majority of depressed persons would continue to be depressed and not subsequently gain (or regain) happiness. Empirically the prediction would be impossible.
One criterion for a good question on this site is that the answer is not obvious. I can't speak for others but I had to pause over your question, so in my book this is a good question.
I used to have a good overview for all the regulating screws of consequentialist theories, but I can't remember right now. So I'll give you some ideas without much structure.
That is not a bad criticism at all. It very much depends on the utilitarian framework in question. Firstly, you'll have to keep in mind that murder is taken as a harm. Certainly it's an indirect harm: so in most cases it will cause suffering to family, friends and so on. But it can also be taken to be a direct harm, even if the person killed doesn't feel anything (imagine painless death while sleeping). If we don't think that's the case then there are conceivable scenarios. There are more factors though. Guilt from the murderer, general changes in society, and so on.
How they get handled depends on the utilitarian framework in question. Let's look at the "aim" and the theory of value behind it first. For total utilitarianism - which looks at the resulting absolute amount of happiness - it's only an issue if and only if negative happiness is a thing. If suffering and happiness aren't taken to be a single scale or if suffering is taken to be a zero then this won't work. For average utilitarianism - which looks at the resulting average happiness - it would still be a problem unless some restrictions apply. There are more options than those, but it should've given you an idea.
Then it also depends which perspective and what object we're evaluating. So if we're agent relative or if we only look at rules instead of acts then this will change things.
Also, the direction your idea takes isn't the only one. I'd advise you to take a look at Parfit's idea of the repugnant conclusion if you're interested in more such issues.
