Secession means you all of a sudden have "foreign relations" .that being part of the United State buffered you from. Could Arizona handle Mexico? Could Texas? Arizona is landlocked. Not good.
Of course it would mean instituting a Foreign Relations Dept. ASAP, best done prior to separating.
I was considering that some of the political work could be done prior to the actual separation, such as negotiating with States and countries, but that would increase the chances of word getting out prematurely.
While it's possible that Mexico would try to expand, Texas does have some modern armament - and the will to use it. Some of the neighboring States might want to get involved on Texas' side too; to stay on Texas' good side, to thwart Mexico, etc.
As was mentioned, Texas has the infrastructure and reserves to survive on its own for a while, at least. Arizona doesn't have the resources Texas does, they might have a harder time of it.
I'm still not convinced that a shooting war would have to occur. What is so important about making someone stay in your organization (Union) if they don't want to belong that you'd kill hundreds, thousands or more?
edited for spelling