If compnay not registered in IL, can they enforce Non compete?

My question involves employment and labor law for the state of: IL and NJ

I Work in IL at a client place through a vendor from IL since June 2008. I am employee of a company based in NJ. this company is not registered in IL and not paying any state taxes and are not deducting any IL state taxes in my payroll instead they are deducting NJ taxes. I became resident of IL since 2009 (moved from NJ to IL since its la long term assignment). I know its wrong but they mentioned they do not want to register in IL and not intend to in future.
(ideally they should be registered and deducting my IL taxes)

in this situation i have been filing returns and paying taxes for IL and NJ both states.

so my question is would this company be able to force non compete agreement in IL if i leave the company to work for vendor directly or client directly?

or can a company enforce non compete in a state where they are not registered or paying corporate taxes to IL?

"How come they can claim that IL is their working/business state if they are getting paid from vendor based in IL for my services to the client when the company is not registered/paying any taxes in state of IL"

I'll rephrase the above part of question,

even if they file suit in NJ, wouldnt the need to show that they are earning income from IL and paying IL state taxes and employeed state taxes are withheld correclty? to show that they have a legit presence in IL.
meaning they'll have to m
make adjustments to my payroll and tax deductions and also pay their part of taxes to IL.

to show that I am competing against my emp. (my emp are consulting company)


if they register in IL, just for the purpose of file suit against me in NJ, how would it hold.
I had asked them to register in IL and do the correct thing, withheld IL taxes instead of making me pay IL and NJ state taxes, which they dont want to do.

also i think being in this situation, how would i be eligible for unemp/disability benifits?
and if i am which state should be the one I filing for, if i need to get unemp/disability?

Please help,

Thanks,