The Hebrew phrase 'chew the cud' means 'raising what has been swallowed.' Coneys and rabbits go through similar motions to ruminants that Linnaeus, the father of modern classification (and a creationist), first classified them as ruminants.
Also, rabbits and hares practice reflection, essentially the same principle as rumination, and do indeed 'raise what has been swallowed.' The food goes through the rabbit and is passed out as a particular type of dropping. These are re-eaten and can nourish the rabbit as they have been partially digested.
In particular, another name for this process is cecotrophy because the material is taken in a pouch at the beginning of the large intestine called the cecum or 'blind gut' (Latin caecus = blind). In the cecum, a process called 'hindgut fermentation' occurs, where bacteria help digest the food by breaking down cellulose into simple sugars. Then the special dropping called a cecotrope, is expelled and re-eaten. This cecotrope is very different from normal feces. Thus cecotrophy is very different from other forms of coprophagy (eating dung) practiced by animals such as pigs and dogs.
The Google Definition of "chew the cud" is: "(of a ruminant animal) further chew partly digested food."
If Hares chew partially digested food, then, by Google Definition, they are Chewing the Cud.