Hello Aspirant,
No , This statement is totally false , any student no matter what college are allowed to get job in government hospital after completing MBBS if the college/University is recognized by government of India. So you are eligible for government jobs in government run hospitals.
Most of the government hospitals in India are short on doctors so getting the job wont be very hard for you as long as you fulfil necessary conditions put forward by them but the salary will not be much . You can find jobs by looking on newspaper or hospitals website.
Hope this helps.
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