Hello Aspirant!

No, absolutely not. It is totally a misconception. Every deserving student is eligible to apply for government jobs irrespective of their college governing body - private or govt.
A private college/school student is equally eligible to a student from a govt. educational institute for the govt. sector jobs.
You only have to qualify the competitive examination related to that job.
There can be some additional criteria, depending on the exam.
But yes, as far as I know, even if one is from a recognized private school or college, he/she can get a govt. job.
Though having a degree from a top govt. institute can sometimes create a good impression during job interviews. But remember, if you have got the relevant skills and knowledge, the job is in your hand!

Hope this answer will help you.