Here are a few possibilities. Take them with a grain of salt, since I know absolutely nothing about you or your neighbor.
1. Are you of different races or ethnic backgrounds? Perhaps she fears your culture or has negative stereotypes and prejudices. Or maybe she thinks you may disapprove of her culture.
2. As you mentioned, maybe it's a religious thing.
3. Maybe she doesn't believe boys and girls of any age should play together. By keeping her boys from even meeting your daughter, she avoids them wanting to set up play dates or go to your house. Maybe she thinks boys and girls playing together always leads to problems of the "Show me yours and I'll show you mine" variety.
4. Maybe she has massive social anxiety, agoraphobia (fear of leaving the house) or a fear of meeting new people
5. Maybe she doesn't speak English well, or has a massive speech impediment that embarrasses her.
6. Maybe she has heard you yell at your children and thinks you're a bad mom (I always worry about my neighbors hearing me when I yell at the kids with the windows open!)
I'm not saying ANY of the above are true and certainly don't think any of the suggestions I made about you are true. I'm just coming up with things someone else might think simply by looking at you, listening to you, or seeing the way you dress, the way you live, when you come and go, etc. I'm sure you're a wonderful mom, person, and neighbor. Just trying to come up with explanations for what certainly seems to be VERY weird behavior.