Can I Get The Blue Plastic Piece If It Breaks Off

Not sure if there are replacements for that piece, but there's a pretty easy work around.  Take the rectangular plastic surround off that covers both spigots (pull down a little and then off to the front).  At that point the spigots just unscrew.  You can replace the broken spigot with a different screw-in spigot of any style if you want.

If you want to keep the existing spigot, just break off and dispose of any/all of the colored plastic from the broken part (as well as the metal pin).  Then make a dowel by cutting a section from a #2 pencil to the width of the black spigot lever.  Place the pencil section in the notched part of the spigot lever.  The pencil will fill the role that the broken blue piece had done.

This will make the spigot work, but it won't be "child-safe" any more, so you may not wish to do this if it's your hot/red lever that broke.  But if the hot side breaks you can always screw it off and replace it with a totally new hot spigot.

The plastic rectangle cover can be replaced or left off as you see fit.