L., your son certainly sees the difference in himself and them - he has a nurse and they don't. He is used to one-on-one attention and now he wants that same attention from the other children. He cannot see the reason that he cannot get his way with them since he does get his way with his nurse. (Her paying attention to him when he demands it.) That's most likely why he hits.
You don't mention if he can talk now. One huge frustration for children is not being able to talk or successfully communicate. If he is behind in speech (which would certainly make sense having had a trach for so long), I hope you are getting him speech therapy. That will help too.
I do think that you have a problem with the school he is in, to be honest. They don't seem to have good handle on appropriate ways to deal with a child hitting. And the moms telling you not to use food in this are spot on - the LAST thing you need is to create a food issue.
I think that it would be a mistake to take this child home away from other children. Staying in a nursing home all this time has given him a babysitter with all old people around him. Having children around him is a good thing - he will learn how to associate with them and have professionals help him learn. There's nothing wrong with you putting him in a daycare instead of staying home with him. But you need the RIGHT daycare with professionals who know what they are doing.
I hope that at some point he will not need a nurse. I have a feeling that he has been coddled some because of his condition - at some point you need to pull back on that and expect him to act like children are supposed to act. Showing love but firmness and consistency is important. Kathleen gives some really good points about showing the other child attention. Also, the teachers should be taking his hands as he hits and saying "Hands are for helping, not hurting. We do NOT hit." Then place all their attention on the other child.
Good luck to you and your son,
Dawn