I don't know much about the EI side but I am certified in Payroll...
If you were being paid as an independent contractor (ie: receiving cheques with no taxes/CPP/EI deductions), then the following will occur:
1. You will not receive a T4 and will need to claim on your own;
2. You will need to claim the money on your taxes and will need to pay taxes on it;
3. If you were not independently paying for CPP/Ei as a self-employed individual, then I am fairly certain you cannot claim EI benefits.
However there is a program in place you can start paying into and it takes (I believe) a year of EI contributions before you can start to claim.
I hope this helps, as I think this is what you were asking about.