Reverse osmosis unit drain clogs occasionally
Most likely you have a plugged flow restrictor.  The flow restrictor regulates the flow of waste water away from the membrane.  If it is plugged for very long, the membrane can become fouled so you might need to replace the membrane and also clean/replace the flow restrictor to restore production.

From the Watts Premier WP-4v manual:



The procedure might be somewhat different if you have a slightly different model.

Regarding the handyman, I have no idea what he thought could be back flushed.