Air carrier pilot reported other aircraft and Flight Attendants told them they observed smoke coming from one of the engines while taxiing for departure. Passengers opened the overwing exits but did not exit the aircraft.

Narrative

While taxiing for departure on XX with both engines running; we saw that the line for departure would likely require more fuel than we had planned. We made the decision to shut down the #1 engine until we were farther up the line. After shutting down the #1 engine ATC changed our taxi instructions several times and resulted in us being 2nd or 3rd for departure instead of the #10-12 that we were expecting. While restarting the #1 engine; we heard several other aircraft calling about smoke from an engine; but we didn't have any fire indications so we didn't think it was us. When ATC told us to hold position we stopped and set the parking brake. Right after we stopped; the Flight Attendants started pushing the call button multiple times; so we decided that it was probably us that had smoke and aborted the start and called for Crash Fire Rescue (CFR) to assist.Passengers opened all 4 overwing exits; so we shut down the #2 engine as well; and prepared for a passenger initiated evacuation. CFR arrived quickly and did not see any signs of smoke or fire; and no passengers exited the aircraft via the overwing exits. Due to the overwing exits being open; we coordinated with ops and CFR to deplane the aircraft via air stairs. Tower closed XX to allow space for CFR and the busses to deplane the passengers; as we were on Taxiway 1 at the intersection of Taxiway 1 and Taxiway 2. After deplaning all passengers and flight attendants; maintenance closed the overwing exits and we were towed back to the terminal. Due to not having any indication of a tailpipe fire we felt behind the curve trying to piece together what was happening from a jumble of radio reports and calls from the flight attendants. Once the aircraft was stopped the event pace accelerated as passengers opened the overwing exits. We were unable to complete checklists until after we were secured for an evacuation due to concerns that passengers would depart the aircraft before we had secured the engines. Due to the rapid onset of an unknown malfunction; and the time compression once passengers started opening exits; we made the aircraft as safe as we could based off of training and followed up with checklists to verify all tasks were completed.