Yes, you can have depressed animals, but not ones that wish they were never born. Not ones that know they don't live in a utopian world. Not ones that can at any moment, hate what they have to do to get by. — schopenhauer1
So let's address these ideas.
First of all, "not wanting to have been born" is stupid. Is it the same as not wanting to continue living? No, one is an act of a desperate or sick person, the other is a whining, childish, poor little me, type of selfishness.
Why do humans wish that they had never been born? Usually because for some reason their life is miserable or beyond that. These reasons are often similar to the ones that make people want to kill themselves.
Why are they miserable? Usually because their lives do not live up to the expectations of what their life should be.
Why does their life not live up to their expectations? There could be many reasons, you found out you were adopted, hubby or wife beats the shit out of them everyday, they don't have the nice house or money that they think they should have, people bully them for wearing different clothes/ black/ yellow/ being short/ fat/ too tall/ red haired/ bald/ gay/ crippled/ liberal/ poor/ etc., they don't get enough likes on farcebook or twatter. You name it, it can be a reason for being miserable, even some of these stupid people that rise to fame and fortune and have everything they could possibly want can be miserable because money, fame, and likes are not everything in life.
Now, why would these people get miserable about these things? Not all but a lot of them are just plain social constructs, nothing at all to do with real life.
Do you think that many of the animals have these problems in their lives? Surely if they don't have the same reasons for not wanting to have been born, or for wanting to kill themselves then they would not think about it.
Why do humans think that they do not live in a utopian type of world? Usually for the same reasons they are miserable. Humans can also think that they live in a less than perfect world because it is no longer the way it was because of our behaviour. We have certainly managed to screw up the world in the last couple of hundred years, but how many people actually think about it?
Do animals have the same problems to think about? They are born, they grow up and learn from their parents, then they go out and live their own lives. Humans look for a nice house to live in and have a family, they build a nest, dig a hole, find a gave and start a family. for them it is utopia.
I doubt that any animal could hate what it has to do to get by, because everything that it does is for that reason. You on the other hand hate what you do because you picked the wrong job, the wrong wife, had too many kids, have a habit that is expensive and cannot afford to pay someone to cut the grass.
When I was a young man I worked for a small local council in the garbage collect department. I spent 3 years working there. The job sucked, there were none of the fancy wheelie bins and special collections like today. But I made a lot of money working there, and we were well respected for doing a very good job. I hated doing it, really hated it, but it served its purpose, paid for a part of my education and I had a lot of great clothes.
May be that is what is missing from the people that hate what they do, they cannot see the benefit of doing it. If there is no benefit, why are they doing it?