Actually - it might be called a center carrier bearing. I know I can buy mine at various vendors - I'm sure one of the big bolt guys will chime in about yours.
As for the grease gun - usually the grease pushes its way out of the four bearing caps (when greasing a u-joint)
Depending on what your putting the grease to, you have to be careful - some of the newer rubber boots that have grease nipples could rupture - so you go slow and watch for the "excrement"
