From their song "Zero" come the lyrics
Emptiness is loneliness
And loneliness is cleanliness
And cleanliness is godliness
And God is empty, just like me
Emptiness = Loneliness?
What makes a person feel "empty"? Is it not having a sense of purpose or meaning? In About a Boy (the awesome movie, not the crap book), Rachel tells Will, "The first time I met you, I thought you were a bit blank." Will had prided himself that "I didn't mean anything, about anything, to anyone." Having no purpose leads to isolation because it is only though our common purposes that we connect with those around us.
Loneliness = Cleanliness?
This is the connection I find most suspect. Billy Corgan is suggesting that uncleanliness comes from others only, that a person can be clean solely by being alone. While it might seem Sartre's line "hell is other people" might support this view, I disagree. I believe Sartre is saying that having to be with other people is punishment, not that being with other people provides the justification for punishment. Sartre says, presupposing uncleanliness, forced socialization is a fitting punishment. Corgan is saying that socialization is the source of the uncleanliness and thus the justification for punishment.
Cleanliness = Godliness?
This is just a rephrasing of the well-known line "cleanliness is next to godliness." Notice "next to," not "is." Being clean gets you closer to God's condition, but there is more to being God than just being clean.
God is empty?
The progression breaks down on the weakness of the middle link. Since other people are not the only source of filth, there is no reason to suppose that God must be isolated to be remain God. He can connect with others through a common purpose. Mormon theology teaches that that purpose is "to bring to pass the immortality and eternal life of man." But even outside Mormon theology, Corgan's assertion that God is empty does not hold up. (Besides, if God is empty "just like me," how lonely are the two of you, anyway?)