A few things

A player has bought a few horses from me and it's annoying that they aren't taken care of. When you sell horses is it best to put them up for private auction? Or will they stay for sale much longer? 
Any other stable owners kick boarders who don't pay? My barn is empty now from players who weren't paying but I feel kinda mean. 

Private auction makes it easier to control who the horse goes to, so it is wiser if you want to ensure that your horse goes to someone you want to have it.

Private auctions taking longer is pretty hit or miss. It depends on who you're selling to, and the horse your selling, as well as you methods for advertising. Public sales always have the ability to go faster since people can grab horses on an impulse and not have to think about it. But As River said Private Auctions allow you to control where your horses go.

As for kicking boarders that don't pay, I do it whenever I remember to check personally I don't feel that bad about it.Those players are using my stalls and they get notified on the horses page that they have low board time so it's easy to see when you might need to pay.