Updating iphone 4s OS 5.1.1 to ANYTHING more current 6.1.1? 7.0?
can I just backup my files and hit update on my phone 


Yes.



Pressing the Update button will cause a jailbroken iPhone to 'go wrong', usually resulting in a reboot loop. Pressing Update on a non-jailbroken iPhone is fine, even if it was previously jailbroken.

To update a jailbroken device, removing the jailbreak…


Backup the device using either iTunes or iCloud.
Restore the device. You can use a previously-downloaded IPSW, or let iTunes download the latest one for you.
Restore from backup. When setting up your iPhone, choose to restore from backup and let iTunes/iCloud restore the backup.