Are Swedish citizens required by law to own a gun?

No, rather the opposite. In Sweden (just like in nearly any European country), firearms are tightly regulated by law, to own any firearms you need a manditory firearms license. The only people who can acquire a firearms license and thus buy and own guns are those who follow these requirements:

- no criminal record

- has proven their proficiency with and knowledge of guns, has proven the ability to safely handle a gun and has proven their knowledge of the respective gun laws through theoretical and practical exams.

- can demonstrate a need/reason to own a firearm, such as having acces to an adequate piece of land to hunt on.