Keeping page ranking on Event pages

The answer (unfortunately standard SEO answer) depends a little bit.

Generally it would make sense to have /june event or even just /event.

The reason it depends a little bit is because it depends on search volumes, search trends, niche and exact keywords. If it is a highly seasonal keyword the rankings shift a lot naturally based on this.

Without more information I would be leaning towards just /event.

"do we have www.website.com/june-event and change the content on the page to match that current information"

If it was me, then yes. 

"do we have www.website.com/june-event and change the content on the page to match that current information"

If it was me, then yes. 

For me, it clearly, keep the page and change the content. The reason is hopefully you will pick up some backlinks to the pages which means that would follow on until next year. It can take an age for google to get round to not only crawling your site but also where these links appear so having a base to start a new year off will stand you in good stead. 

The other thing to make sure you do is to make sure you add schema and make sure you change it for each new season. Google are tending to use schema more and more and for events such as this it can really help.