Yes it support iOS 7 as I used it on my iPhone 5 yesterday. Basically here is what you have to do:
First I would recommend making a backup of your phone to both iCloud and iTunes
Download it from here:
http://www.icopybot.com/itunes-backup-manager.htm
Make a backup of your phone in iTunes.
Open iBackup bot and on the left choose the backup that you just made. Right click it and make a duplicate and save that anywhere you want just in case something happens.
On the left, click System Files -> Home Domain -> Library -> SMS
then click the sms.db and from the top menu export this file to your desktop.
Now that you have the sms.db file, restore/clean install your phone in iTunes to iOS 7.
Once it is done make a backup of the freshly restore iPhone through iTunes.
Go to iBackup bot, close it, and then reopen it. Now click on your new fresh iOS7 installed backup on the left. Go to the same place as before (System Files -> Home Domain -> Library -> SMS)
Now your gonna want to click import on the top bar and choose the sms.db file that you saved earlier. When it asks you to replace click yes. Once you have successfuly put the sms.db file into the backup then close iBackup Bot and go into iTunes.
Once your in iTunes click restore backup and choose the new backup that you just put the sms.db file in.
Let it restore and your done

You now have a fresh install of iOS 7 with all your old text messages.