What are the most important qualities that a technical support engineer must have ?

He should have troubleshooting skills to diagnose problem at earliest.

Customer are always attracted to support engineers who give priority to them and diagnose the fault quickly.

Furthermore give priority to get the issue solved is also one important quality.

Communication, Technical skills,  Problem solving skills, Time management, Learning new technologies

Patience : ) .

There will be too many issues at job and technical problems beside working with usually people who have no knowledge about computers.

solving problems 

good knowlege of technologies 

good troubleshooting and fix problems

good communication skills 

team player