After my employment ends, can I keep working without pay at my current research center, on independent research?
Of course anyone can be an independent researcher.  Just do it.  But that's not what you're really asking.  The question you're really asking is this:


  How can I convince my former supervisor to let me continue using their equipment and resources after my current contract ends?


And the only reasonable answer that question is to turn in on its head.  What's in it for them?  How would they benefit from giving you an office and access to their equipment?  Is the payoff for them worth more to them than the resources that you would expend?  The fact that the project is good for you isn't enough.

There's an even more basic question: Do they have resources that they can give you access to?  The equipment that you want access to may be tied to the specific project.  They may not be able to give you access.  But the only person that can answer that question is your former supervisor.



Some updates in response to the updated question:


  However , i need my supervisor to draw public/scientific attention , make a paper with him out of it etc.


No, you don't.  You can write and submit a paper yourself.  You can publicize it yourself.  You can ask your former supervisor to help you publicize it; if your supervisor was your advisor and not just your boss, they'll be happy to help you, because your success makes them look good.


  I am afraid that he could simply use my idea with the engineers that still have a contract there.


You don't trust your supervisor.  So don't work with him.  Your mistrust is probably misplaced, but that's your call. To quote Howard Aiken: "Don't worry about people stealing your ideas. If your ideas are any good, you'll have to ram them down people's throats."


  of course my supervisor be the one that would present it and his name next to mine


Why "of course"?  Just do it yourself.


  i dont want my supervisor to demand things on the project.


No, that's not how it works. If you want your advisor to put his name on something, you must give him partial responsibility for that something.  That's what coauthorship means.


  I am thinking of proposing to him a complete solution of exactly what i have in mind , let me work on it with my time , without any fundings , but he ll be the one to present it and draw attention on it.


In other words, you want to do all the fun stuff, using your supervisor's resources, and leave the hard work of writing and presenting to your supervisor, but not allow them any actual input, and then claim credit for yourself.

That's just not going to happen.  If you want something from someone else — working space, laptops, Internet connection, help with writing and presentation, anything — you have to offer them something in return.  Either offer to make your supervisor a full partner, including veto power, or you do the work yourself, including all the writing and presentation, with your own equipment, on your own time.

Having the project on your CV will open more doors if you do it without your supervisor.