Company not allowing me to cancel membership

Undervalued

John_ said:

It looks as though you are asking them to waive the contract which you signed, and are claiming that you have a good reason for asking, but then refusing to give them any evidence to validate what you are telling them, is that correct?

If so, why do you think that they should let you out of the agreement? From their point of view, you may be lying to them.

^^^ Exactly this.
If the contract allows an opt out in the event of illness (frankly unlikely) then it is perfectly reasonable for them to want proof. That could be in the form of a doctor's letter confirming that the OP was too unwell to work but without giving personal medical information. However the OP would have to pay the doctor to provide that.