Along with current chairman Jed McCrory and general manager Steve Murrall, Hooper and Rice were added to the board of directors in February following the takeover by Seebeck 87 Limited.
Following the initial transition period the pair have been spending less and less time at the County Ground in recent weeks and have now returned to concentrate on their own outside business interests.
"I would personally like to thank Gary and Callum," McCrory told the club's official website.
"We have had a very testing three months and their efforts have been much appreciated during this period.
"Callum has stepped down to concentrate on his own business interests but has enjoyed his short time involved at Swindon. Gary will also be focusing his efforts on his own business.
"We have a solid foundation to take the club forward and now it is the time to strengthen the board with the skills required for longer term sustainability and success."
A club statement added: "Swindon Town Football Club would like to thank both Gary and Callum for helping with the transition period and wish them all the very best in their future endeavours."