It is a little ambiguous, because United's policy is that if you are too sick to fly, you pay for the change fee now, and then request a refund on the change fee along with doctor's note saying travel was not recommended on the original date.
https://www.united.com/web/en-US/con...ds/refund.aspx
In the event that your travel plans change as a result of illness or jury duty, you will be required to pay the applicable change fee at the time of that change. Once this change fee is applied, you may submit a request to have the fee refunded.
However, with Basic Economy there is no change fee -- the ticket cannot be voluntarily changed. To get it changed -- fee or no fee -- you'd have to show ahead of time that you are too sick to fly. And I'm not sure there's a procedure in place for that -- only the refunds department evaluates doctor's notes, not the reservations department.
Only if you get a physician to state that you can't travel during the entire period of validity for the ticket -- typically one year -- will they refund you.
So unless someone here has had experience with or sees a report of someone trying to change a BE ticket changed due to illness, it's all speculation.