Hi A.,
I want to start by saying that if he is not talking a lot at home, he is probably not doing it at day care either. Day care centers have a habit of making things sound better then they are (I have worked at a few 4+ star daycares, and quit because of how the children were treated and how much they lied to tha parents, I now watch kids in my home to keep at least a few out of the centers.) Also, he probably hears "no" constantly at the center.
I try to refrain from saying "no" to children. More positive phrases such as, "That's not OK," or "walk away" are more effective for most children. Instead of "no" try telling him what you dont want him doing. For example, say he is climbing on something, try saying, "Lets put our feet on the ground." If he is touching something he shouldnt be touching say, "that's not ok, lets walk away." If he hits say "Lets use our nice touches" and so on. You would be AMAZED at how well this works. I remember once I was at the mall play area with a 1.5yr old I used to watch, he was wanting to climb up the slide so I told him that was no ok, and asked him to walk away. A mom next to me said, "That will never work," and was AMAZED when the child walked away from the slide. If they refuse to walk away say "would you like me to help you walk away?" and then take his hand and do just that, help him walk away. Not only does this prevent children from loving the simple word, "no" but it teaches them more vocabulary.
Something to try for more vocabulary is to also have conversations with him. Tell him what you are doing and why- like when you are changing his diaper say "Im changing your diaper because yours is wet and a new diaper will feel much better." When you are making lunch or dinner tell him what you're doing. When you give him juice say "here is your juice" and so on. So many parents get into the habit of mimiking their children's baby words and that does not help. If he has little names for things, repeat him with the real name for it. If he wants something say to him, "Use your words" and then help him by saying, "you would like something to drink? Say, Juice please." and then give him the juice.
I am currently watching a two year old who does not talk much. He has been here for about a month and is really starting to try to use words. when ever he grunts and points I ask him to use his words. He is really starting to try. When he does use his words I make a really big deal out of it and he is quite proud with himself.
Good luck!