There is no reason that welfare recipients should be able to send their kids to anything but a public school. My parents who work for a living did not send me to private school, so they should not have to pay for someone else to do so. It would be wrong in every way.
Yes, recipients of state-supported welfare programs should be required to send their children to public schools, because if they are homeschooled, they do not have the tools to home school effectively, and the children will probably go uneducated. If they send their children to private school, they have enough money that they do not need welfare.
Parents are required to send their children to school although each district does allow home schooling. Home schooling could be a viable option for some people who receive assistance, so to say they should be forced into public school is entirely unnecessary. If you're asking if it is wrong for these parents to send their children to private school I'm going to have to laugh, because people who actually qualify for what people consider welfare can't afford those tuition prices to start with.
I do not think that people who get welfare from the government should have to be forced to send their kids to public schools. I think that if they are able to teach their kids at home, etc, then they should have the right to choose. It is more important to make sure these people deserve the benefits though.