I am a non-U.S. citizen with authorization to work in the U.S. I have recently earned a teacher’s certification and a Master’s dergree in special education from a U.S. higher education institution. I applied for a teacher position at one of Texas ISDs, went through the interviewing process, and got a call from HR verbally offering me the position. After they learned I am not a US citizen, they changed their mind. Consequently, they asked the school principal to look for another applicant, and they deactivated my online application to avoid principals from contacting me in the future. They clearly stated to me that they are no longer going to offer me the position since hundreds of US citizens have applied for the same position. I lost a lot of time and went through huge emotional pain!
This appears to be a labor law question, not an immigration question. However, whether you are “employable” under immigration law depends on what kind of employment authorization you have–i.e. on what basis you obtained permission to work. Many work authorizations are tied to a specific employer, while others are valid for any type of work. For example, a lawful resident (green card holder) may work anywhere and in any job. A person with an H-1B visa is restricted to the employer who petitioned for him and at site specified in the visa petition.