how do you identify your one specific record?
If it's by a unique primary key, then you can use conditional formatting to 'hide' the button by setting its text and fill colour to the background, and you might want to turn off the 'Change to hand cursor over button'
You should also consider setting the first line of the script to check the record iD and not proceed if 'accidentally' clicked on another record
What is the purpose of having the button visible on only one particular record? I can think of a number of reasons and many are better handled by other methods such as passing a script parameter or placing the button in the header and/or footer layout part.
For ways to hide a layout object, see this demo file: https://www.dropbox.com/s/8levaz6deiyzjr8/LayoutObjectVisibilityControlDemo.fp7
It's in the older file format so FileMaker 12 users will need to open it from the File menu to get copy converted to the .FMP12 format.