Do they put the name of the company that is hiring you in your passport when you apply for the non-B? The reason I ask is because I have a school which I am teaching for and they gave me my paperwork to go get the non-B (currently on Ed-visa). However, a much bigger and better job has been offered to me (non-school), and so I don't want the school job anymore and do want the new job. Can I use the school's paperwork to go get the non-B, then quit the school (before WP is assigned), and then use that new non-B stamp to get a WP for the new job?

You got the non B based on working for the school you first went to.

They're not that lapse here.

The shorter answer is I don't know, but i wouldnt.

Answer: NO you cannot unless you want to face charges and possible deportation.

Opinion: I don't want to be rude but I think you already know the answer to your question or at least this new "better" job should know and it is people like yourself who operate illegally here in Thailand for your own prosperity ignoring the fact that this teaching job sounds like they are treating you will all due respect yet you are willing to just trash them that make the laws difficult for us who live here and want to keep legal.

If you want to do the right thing you should tell the school that you do not wish to work for them any more, decline the paper work they gave you and STAY LEGAL. Then you ask the new job for the proper paperwork and continue on.

No name of any company or school is put in your passport. Only in a Work Permit.

I see you opend up your own topic, you got my answer in your other topic on this.