Exactly what are button-type signs? I work at the City Hall here and can get pretty much any sign made that they have the metal form for.

"Button-Type" signs are also known as 'Cats Eye" or "Marble" signs.
For the BGS (Big Green Signs), they are mounted in the removable letters. So instead of the whole sign being reflective, only the cats-eyes are. Sometimes they're still used (in the long run, saves the DOT money, don't have to light-up the signs with catseyes.)
For normal signs, they got phased out looong ago, because of cost.

For a good exmple, go here.

They're not too easy to make nowadays, with the pre-fab aluminum blanks, since it requires two sheets, one cut out for the eyes, the other for backing.. But, they look darn cool.

Me.