Yes, 4 yo can be depressed. Keep in mind a depressed mood can be caused by any number of things. Is he getting enough sleep? Does he eat enough to maintain his weight. Does he enjoy doing some things even tho he hates other things? What is important is the total picture.
It is common for kids, even at this age, to say that they hate everything. He could've picked up this phrase at school and is trying it out to see what sort of reaction he'll get. Running to his room crying does indicate that he's upset but not necessarily that he's depressed. My grandkids run to their room crying when they're tired and aren't allowed to do something that they want to do or are told to do something that they don't want to do They are frustrated or even angry. Running to their room to cry is one healthy way to express their feelings.
He may be reacting to the serious way that you're reacting to his statement. He could be feeling that if you're taking him seriously something may be wrong.
Another aspect to consider is how much approval is he getting at home and at school and what sort of discipline is used at both places. Is the discipline consistent and firm but respectful of him as a child? Is it given in a calm, reasonable, child accepting manner? Is the tone of voice neutral or tending towards anger?
How are you and your husband getting along? Is the household experiencing stress or tension? Is it chaotic? Even tho everything is OK is there serious discussion that could feel like negative energy to him? I remember feeling anxious at a very young age when the adults at family dinners argued about politics.
It does sound like something is going on to make him unhappy when you say he cries when you pick him up and he wants Dad. Could you be trying too hard to soothe him so that he'll be happy? Perhaps talking him out of being upset? I see this often. It's something like "don't cry. You're alright." when he's not feeling alright. It sometimes helps to say, "Moms got you now. You're OK." A subtle difference but one that some kids pick up on and others don't.
Are you feeling confident and/or showing him confidence in your ability as a parent? When parents try "too hard" make their child happy this can cause them to feel insecure. Children depend on parents knowing what is best for them.
My suggestion is that you take a look at what is happening in these areas at school and at home and perhaps modify some of what you're doing. I wouldn't try to talk to him while he's upset. Let him voice his feelings and comfort him. Then, later, when he's playing, try asking some questions while you're playing with him. Be casual so that he won't feel that what he says is important.
You can also read books about feelings with him and leave plenty of time spaces and ask questions so that he's comfortable making comments about what you're reading. If the book is about a child, ask him what he thinks the child might feel like before you turn the page to read what he did feel like. Use this opportunity to discuss these situations as they might relate to him but use the books character as the focus.
Tell him stories about situations that you want to know about. For example tell him about a little boy who hates school. Ask your son why he thinks this boy hates school. Ask him what he thinks could be done so that this boy would like school. Do the same thing with anything you want to know more about what he thinks.
Do remember that saying, "I hate school It's boring." does not necessarily mean it's boring. Boring is a catch all word for dissatisfaction. Go one step further and ask why it's boring. If he can't think of anything to say, make some suggestions such as "is it boring because you have to stay in your seat?" If his answer is yes or seems to be in that direction ask him what happens when he doesn't stay in his seat?
I suggest that you also observe in the classroom and on the playground if this is at all possible. Do this several times. Make an appointment with the teacher and ask her what she thinks is going on.
I suspect your son is the only child with a parent working a rotating schedule. He may not understand the schedule and feels apprehensive or unsure when he'll see his Dad. My father, a fireman, handled this with us by marking a calendar so that we could look at it and know what shift he was working. He used a color code. If the week was colored in blue we could look at the symbol on the side and know that blue meant he was working from 6am-6pm. He worked something like 4 days on and 4 days off. His days off were not colored. We also talked often about his schedule.
Do you work the same days and same hours all the time? Remind him the day before that you'll be working tomorrow but Dad will be here.
In fact, talk with him about what is happening as time goes along. As adults we pick up on cues about the next activity but kids haven't learned how to do this yet.
If the school has a counselor it may help to talk with them. They've had training and experience in what to expect from children at various ages. It sounds to me, based on the little bit you've been able to describe in a small space that he's feeling anxious but is not necessarily depressed.
