My short answer is no, but read on for a more detailed explanation.
Schools do want teachers who their students will be able to understand. This encompasses word choice, speed, use of idioms and general communication skills, though — much more than just your accent. Students are learners, and I have taught very few who would be able to easily distinguish native English speakers from different countries. They might be able to understand Brits or Americans better in general, but it is relatively rare for learners to be able to identify people’s accents on the spot. Schools may also say, as one justification for hiring native speakers as opposed to local teachers with language degrees, that they want natural accents…but ironically one of the first things that TEFL courses teach is how to grade their language. Also, EFL teachers to have a tendency to develop a “TEFL dialect” over time. Nonetheless, native speakers have the huge advantage of being in demand.
Schools may have a preference as to the country your passport is from because of the difficulty or ease of obtaining a work permit. I am thinking primarily of the EU, where it is much easier for those with UK or Irish passports to work legally. There may be special programs designed for citizens of a certain country which allow those nationals to work…but then this requirement would be connected to eligibility for a program, not to employment at a language school. This situation may also mean that people who are in fact native speakers of English but hold passports from countries that are not generally considered English speaking countries may face more hurdles getting work.
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Students have all kinds of ways they’d like their teachers to be, and the nature of language schools — where students are essentially clients or customers — means that schools may cater to this. If students want blond-haired, blue-eyed teachers from the US…that’s what the schools look for, for better or worse. If they request proper British accents, that’s what they get. And schools may have large enough pools of applicants that they can essentially hand pick people…but I would question the quality of an organization that would base an employment decision on something so superficial.