Can I Change the Layout of the Print Screen?

You cannot change the layout of the print screen.

When you print the record details screen, its layout would be the same as the one that you set in the "Form" setting of the app.
Only one form can be created per app.

As an alternative, you can use the integration services that other companies provide if you have kintone Standard license.
Find out if there is any desired service available for you by searching the Extensions page with a keyword such as "layout".