When I first read your post I assumed your kids were elementary aged children where deliberately not saying "good morning" to a teacher is indeed very rude and that would definitely need to be addressed. However, after reading your posts later on, it appears that the children were preschool ("nursery school") age which would put them at around 3-4 and that is a wholly different age group with very different expectations, even age 3 is very different from age 4. So, if it were my kid and they were three and didn't want to greet their teacher at the door but everything else regarding their behavior/interactions at school were fine, I would keep encouraging them to say "good morning" but also try suggesting even just a high-five or a wave Something to get them to acknowledge their teacher. If they don't do it right away, that's fine. It's something to work toward because, again, they are still only three. I would rather my kid greet their teacher because they are confident and comfortable rather than afraid of being punished. I would also ask them why they don't want to say good morning and kind of figure out what is going on there. One approach that I've found very useful and effective for children this age (especially for teaching about manners and how to act in various social interactions) is "pretend play." Basically, you play with your children using dolls or stuffed animals, GI Joes, etc., and act out the scenario you're trying to teach. This gives you the chance to
show the right and wrong way to act in a certain situation. It also gives your child the chance to practice the correct behavior. So, your situation with the teacher for example: you would act out (with the toys) what your child is doing but in an exaggerated way. One doll is the teacher, one your child. Teacher says "good morning!" Child "No! I won't say good morning! I don't like you!" (or something like that) Teacher starts to cry and says "that hurt my feelings! Wah!" What this does is shows the child what they are doing but in their own language and then you can talk about what happened in that scenario. After that, act out the behavior that you want to encourage but exaggerated. Kid "good morning Teacher!" Teacher "Thank you for saying good morning! That made me very happy!" Everyone says "yay!" Then talk about what happened and why it was a better choice to say good morning. I know it seems kind of silly but I've found this to be very effective with little kids. It also gives you a chance, as a parent, to show/talk about making the right or "good" choices and that how we act effects others.
If it was my kid and they were older (let's say 10), I would have used a different strategy. Example: Son won't say good morning to his teacher. I tell him he's being rude and disrespectful and that he needs to say good morning. He says no, I won't. I say "if you won't be polite to your teacher and say good morning I am taking away [favorite toy/privilege]. When you decide to say good morning then you can have it back." Every time they ask for the item/privilege I calmly remind them they can have it back once they start saying good morning to their teacher. It might take a day or two, maybe a week if they're being stubborn, but they always give in. This strategy is useful because 1) you avoid a power struggle and 2) you give your kid the chance to make the right choice. Once they make the right choice (you have to trust that they will), give the item/privilege back and acknowledge them with a bit of praise ("I knew you would make the right choice.") and move on.
All in all, I think using a belt on a three/four year old (or hypothetical ten year old) to make them say good morning is an extreme over-reaction to the situation and could be handled in a different way that, while more involved, is more effective, especially long-term.
You said that maybe my kids "were different" but they're really not. I mostly spent a lot of time and effort with them when they were little (which is the proper thing for me to do, I am their mother after all) and now I have very few behavior issues with them now that they're older and the ones that do pop-up are minor and simple to deal with. I largely agree with everything
@PurpleUrkel wrote (minus the personal attacks).
Anyway, the book I always recommend new parents is
Secrets of the Baby Whisperer and then
Secrets of the Baby Whisperer for Toddlers.