What does Shimano recommend to put into their brand as far as maintaining their worm gears?

Shimano would tell you to use their Permalub product.  I've used light oil, ATF, and dry lubricant with success.  Light oil is probably the easiest to maintain.  Anything heavy or greasy seems to attract dirt and debris.

2 hours ago, J Francho said:

Shimano would tell you to use their Permalub product.  I've used light oil, ATF, and dry lubricant with success.  Light oil is probably the easiest to maintain.  Anything heavy or greasy seems to attract dirt and debris.

Is that permalube the same thing as their “drag star grease” ? The only thing I’ve seen available on their website has been the bantam oil and the star drag grease. I have both just not sure which one should go into the worm gear.

No, Permalub is way different.  You're best off calling them directly.

Some blue marine grease or super lube would work. I prefer a greased worm shaft over oiled. 

The high end dry lube for motorcycle chains seems to work well for me and does not collect dirt.

I'm not sure what Shimano recommend but I alway use grease on disengaging  level wind worm gear. I don't apply a lot just enough to cover inside groove. Some might use heavy lube 90w but I found I have to apply lube more often. With non disengaging type reel always light lube.

BreakFree CLP is good.

Tom

From day one I have only used ReelX on my worm gears and it has served me well, but not what Shimano has recommended.

ReelX is a very good product for exposed mechanicals.  I also like it for roller guide bearings on spinning reels.  They have to be the most neglected bearing.  I forgot about that stuff - thanks for adding it to the list!

15 hours ago, WRB said:

BreakFree CLP is good.

Tom

 BreakFree CLP is actualy what people think WD-40 is, if that makes any sense.  I love the stuff and along with white lithium grease and ATF fluid, forms the holy trinity of "do all lubricants".