It should drip out if it's working correctly.

I would of said a slideway oil, however linear slides will work with different oils/greases, and probably won't benefit as much from a dedicated slideway oil. I'd have a read of some linear rail manuals (Hiwin's catalogue has info on lubrication) to see what they recommend.
You'll need an oil of some form, as the type of lube system you describe won't work with grease.