Unlikely, since you'd risk that the hat start to slump too far forward or backward on his head. It would have been secured in place just where they wanted it on his head, with the built in ability to remove it when necessary.
All this stuff is designed to be optimal for constant repeated use, consistent show, and maintenance access. Large Snaps (think Whopper Poppers) are popular because they're hard to undo by accident but easy to undo on purpose, don't wear out like Velcro, and can be placed precisely and integrated into a wide variety of textiles and skins. They're also easy to replace and can launder with costumes.
Magnets are now used too in some cases, but I doubt they would have been back in Buzzy's day, at least not for his hat.