Are there any freely available plugins or integrations for JIRA to write test cases?
There are several paid options, such as Zephyr, Xray, and Test Management for JIRA. We use Zephyr at my company. All of these plugins have free trials available, but I'm not aware of any that are free.
I'd recommend browsing the Atlassian Marketplace and filtering by "test" and "free" to see if any of the plugins fit what you need: https://marketplace.atlassian.com/search?moreFilters=free&query=test
Just be aware that some of the "free" plugins require an outside subscription to use.