If any single state had a shot at being a viable nation in and of itself, I think Texas or Alaska would be it.
Their oil resources would be a nice nest egg to start with until they could develop their other industries (ranching and agriculture) to the point where they could be economically independent of the US.
Texas' position on the Mexican border as well as their control of the beginning of the I-35 corridor gives them the ability to tax goods coming through their territory. The port of Houston is also a huge source of revenue.
Texas is a tax donor state, meaning it gives more to the federal government than it receives. So, while Texas would have to raise it's own taxes in order to meet its current budget, the individual tax burden on the people living in Texas would not change. Unlike Louisiana, an independent Texas could pay for itself.
In terms of being able to defend itself, Texas would automatically assume control over seven Air Force bases, seven Army bases, three Navy bases, and a Coast Guard base. I'm sure the US would withdraw its forces, but it can't move those installations. I don't know what the US military would do with respect to its Texas members, but the Texas National Guard (Air Force and Army components) would all fall under the control of the new country. I don't have an exact number, but I know the Texas Guard is HUGE.
It couldn't stand up to the rest of the US, but it would be big enough such that your average US politician would think twice about invading.
I obviously do not want to see the US divided, but I would find it absolutely hilarious if Texas did secede and even more so if they succeeded as a nation.