[A2A] In US, no one cares about a no-objection certificate from a previous employer (in this case, Indian Government) especially since you are applying to be student and not for a job. So, I don't think you should have a problem getting a US student visa even if you don't have an NOC from your employer. Just be honest about your current job and tell about your intentions of quitting the job if they ask you.
I am not sure about permission from the government to apply for a visa. Sorry.