Can I go threw a different insurance company rather than my employment to pay for my child's health insurance when it would take half of my paycheck to cover the child threw my place of employment?

I don't see where your support obligation would require a specific health insurance policy as long as they are covered. But it might. You need to read the support order.

However, I can't imagine where a private policy would be cheaper than an employer sponsored group policy.

I hope if you're really losing 1/2 your paycheck to insurance that it's a really good policy! My biggest concern in changing insurance would be the co-parent not being satisfied with the coverage. Remember, the lower the monthly cost the more everything else costs - copays, deductables, drugs etc. If the other parent has to shell out more $$$ because your private insurance isn't covering everything, they might ask the court to revisit the topic - which could cost you more in the long run.