You do not need to go to Asia for your first teaching gig.
The difficulty with Spain would be connected to their membership in the EU. That is why many employers there would want you to have an EU passoprt, it just makes it easier to get you legal.
I've been in the Czech Republic for four years and I can tell you that being legal for EU citizens here is a breeze compared to a lot of non-EU citizens. Its all about the bureaucracy.
I've had many colleagues, both EU and non-EU citizens who have worked in other EU countries and beyond a doubt the bureaucracies of the EU countries are set up to favour citizens of other EU countries.
It may also be a matter of what the country legally expects from employers within it. It could well be the Spanish law stipulates that English teachers must come from within the EU. I'm told that is the case with Belgium.
Am I assuming correctly that you have checked the websites of the embassies of the countries you wish to go to? They should all have some information about requirements to work within their borders and a general overview of their labour codes regarding foreigners working there.