Can I create a button that will update a field of every record in a report?

Can I assume that you have tried Grid Edit and its not good enough for your needs?  Are you aware that you can edit the first cell and then highlight the whole column header and then right click to fill down.

Are you also aware of "search and replace" in the more .. menu.

I just want to get the easy solutions out of the way before we get into more complicated suggestions.

I am aware of these but I'm hoping to set this up for less technical users. Everyone loves a button. 

OK, so there are ways to mass update a set of records with a button, but there needs to be a way to identify the "set".  Simply being on a random report with a bunch of simple or perhaps very complex filters is not enough to be able to call an API to do a mass update.  But if there are a a limited number of standard filters for the report there are some approaches, but there is nothing simple.

I usually arrive at this set of records via a batch search, as set up here: 

https://community.quickbase.com/quickbase/topics/search-bar-increase-character-limit-past-250

Would this be enough to work with natively? 

Yes! It would.

I did not follow, could you break this down to a list of steps that one would need to do to accomplish this?

You will need to go through that thread and follow the steps and ask specific questions, or else contact me via my website QuickBaseCoach.com for one on one assistance