I think this is a Siberian phenomenon only. Kolchak apparently had units that were "
druzhina of the Holy Cross". Druzhina being a hard word to translate. Not regiment - more militia or detachment. A banner for one lot can be seen at
http://www.vexillographia.ru/russia/index.htm . Go to Flags of White Forces and it is the green one with the cross, crescent and stars.
The situation out in Siberia after 1920 is harder to get a handle on. Dieterikhs was a bit of a religious nut-case, and it wouldn't surprise me at all to see units like this under his command.
I've not seen reference to units like this in the South or West. That isn't to say that they didn't exist, just that they either didn't last very long or had another name as well.
Another option is that some weren't White at all, but Green.