Dzemaili, 20, is already captain of Zurich and featured in Switzerland's World Cup squad last summer as well as the friendly against Germany on Wednesday.
He has agreed a two-year contract while 23-year-old Cid, a former France Under-19 international, has committed to four years.
Bolton boss Sam Allardyce told the club's official website: "Blerim is a fantastic player, who has played for the best team in Switzerland for the past three years. For someone so young, he has a wealth of experience.
"At the tender age of 20, he is the skipper of FC Zurich and is expected to become a regular international for Switzerland in the years ahead.
"He has hardly missed a game for FC Zurich since he started playing for them at the age of 17. I am excited by the prospect of working with Blerim next season."
He added: "We have been monitoring Gerald for quite some time and our scouts have rated him as one of the best young defenders in France."