Mars does have an atmosphere, but is there enough atmosphere to glide, shed speed and land if there was a runway? What would this require from the shuttle, in terms of aerodynamics?
The essential problem is to build something strong enough to withstand the reentry from orbit, with a big enough wing surface compared to its mass to glide while moving slowly enough to land, and with a low enough mass to be more economical than a propulsive landing.
As @Dragongeek has said, lift is very roughly proportional to air density and wing area. Now the air density on Mars is roughly 1/50 of that on Earth (from here) and the gravity is roughly 1/3, so we need about 17 times as much wing area per kilogram of mass as the Space Shuttle for the same landing velocity. We might be able to bump up the landing velocity a bit, but the shuttle already lands pretty hard and fast.
What that tells us is that, as it comes in to land, the Mars shuttle would have to look a lot more like a glider than like a space shuttle, with huge thin wings. On the other hand, a vehicle that shape seems very very poorly adapted for reentry. Mars reentry is a bit easier than Earth because the escape velocity is lower, but even so, it's hard to imagine huge thin wings not being ripped off, or simply melting, as well as creating so much drag that the deceleration might kill the crew.
So the only way I could see this working is a variable geometry vehicle that could reenter with wings folded in some way and then extend them for the final stages of landing. The engineering and materials challenges would be formidable, but I don't see any reason why this is necessarily impossible.
Finally, though, we get back to the question of why? Folding wings for the final stages of reentry and an undercarriage to land on would add a huge amount of mass, complexity and potential for error, to do basically the same job as either parachutes or rockets.