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......Also, about there not being posts, I believe that was the point of a Leo Quan. To install the bridge easily without drilling holes or anything
Not really. This piece is a combination of a tail piece and a bridge all in one. Google Paul Reed Smith guitars and take a look at some of their guitars. Some have a stop piece, others have this type of wrap around bridge.
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. Is there any place I could buy the post for it to sit on/in my guitar? One of my friends has a Tune-o-Matic style bridge, and his just sits on the wood, I don't see any posts. Are they supposed to go in the guitar?
If your friends tune-a-matic bridge is set into a piece of wood that sets on the top, it is referred to as a floating bridge. These are found on arch-top guitars, mostly jazz type guitars. Or, similar to a Gretch guitar like th kind Brian Setzer of the Stray Cats and Brian Setzer Orchestra plays.
This Leo Quan does not fit onto a strat style without lots of work and drilling. The posts used for this are usually about 3/8" in diameter, if not more. Compared to the smaller posts the actual bridge sits on.
The Red Ibanez has the bridge on a piece of rosewood. It is the Floating bridge style.
The middle pic is a Paul Reed Smith. It has the wrap-around bridge and tail piece (or stop bar) in one piece, similar to the Leo Quan.
The last one is a Les Paul, The stop bar is separate from the bridge. So, your friend would have to calculate and measure very carefully the precise point to drill and place the posts, other wise intonation would be near impossible. And from the pic you showed of your current guitar, I personally don't think it would be worthwhile to invest money into that.
Sorry, just my opinion form working on guitars for the last 15 years.