Yes, the 529 distribution is tax-free if you spent it on private school tuition in the year you withdrew it, and no it does not matter that you live in Texas and used a Nebraska plan.
1099-Qs are informational, if you know that it isn't taxable, you don't need to enter it.
Here is some information you might find helpful: Guide to IRS Form 1099-Q: Payments from Qualified Education Programs
