There is something very wrong here. I am a former preschool teacher (before I had kids) and I have a master's in education, and this school and this teacher sound like they do not have an understanding of developmentally appropriate practice.
First, requiring children to do an "entry task" that is academic is NOT appropriate for four year-olds. Yes, some children this age enjoy "worksheets", but it is not developmentally appropriate for 4 year-olds to be spending time doing academic tasks. Many, many children are not ready for this type of activity, and it's very likely that your son is still within range of what is typical for his age. (Of course I have never met him, but really, it's fine for a four year-old to be uninterested in letters!!!! If you are concerned at all about his fine motor skills, that;s something you can assess in other ways besides letter writing...)
So, not only is it inappropriate to require this of four year-olds, it is DEFINITELY not okay to isolate him when he does not want to do the work! There has been lots and lots of research that shows that preschool aged children learn through play, NOT academic activities such as worksheets that isolate skills. Research has shown that children who attend preschools that focus on developmentally appropriate play and avoid academics have better school performance in elementary school. The most important things that preschoolers should be learning and practicing are building friendships, getting along with peers, conflict resolution skills, building relationships with adults outside their home, creativity, problem-solving, building oral language, enjoying books and stories, etc. How can a child accomplish these tasks if they are spending their time doing worksheets and sitting by themselves?
I am saddened by your post. The things the teacher said to you are very inappropriate and unprofessional. And I just can't believe that the teacher forces every child to hug her. That is just awful! How can we teach children to have control over their own bodies and say no to touching they don't like when their teacher whom they are supposed to trust forces them to hug her? That is inexcusable, and no adult should EVER force a child to hug.
I think you are certainly not being a "nagging" mom. You are listening to your instincts and protecting your son. If it were me, I would probably start looking for another school. I sincerely wish you the best of luck.