I very much liked FakeMod's and LukeLYU's responses. This is an extension and generalisation.
Let's stick to idealised shapes and assume no additional energy is input beyond that required to start the motion.
When you raise the square to stand on its corner, it won't automatically continue to roll. It has to use some of its energy to move past that point.This is because the corner is a special point corresponding to a point of inflexion (metastable equilibrium) in the PE. This potential curve will have a series of such points as the square rolls. By symmetry, for a uniform body, all these points of inflexion are equivalent in the sense that shape of the potential curve "close to" the local extremum is the same.
Similarly for the flat surfaces to continue to roll energy has to be expended to pass on and as it passes through each point it will loose energy (only here the potential loss is deeper) The PE will thus decrease, slowing and eventually stopping the square from moving further.
For a circle there is no such point of inflexion. All points on the surface are equivalent and since there is no special point, there is no extremum. 621
Once started rolling it will continue rolling without any additional applied force. (<PERSON> 1)
We can extend this argument to polygons. Champfer off each corner to an equal amount, then not only does each vertex correspond to a point of inflexion, so too do each of the new flat surfaces that we have created. Again by symmetry, but at each point of inflexion energy is lost, but it will loose it quicker.
[If you continue this process ad nauseum however you do not get a circle! Since the group representing the translations of a circle is continuous, SO(2), and that representing those of an n-gon is finite (D2n), you do not get the same dynamics.]
By exploiting similar arguments we can consider n-dimensional analogues, but we have to be careful: for example a cylinder in 3D has a preferred axis (its length) and is not a analogue of a circle in this sense, however a 3-sphere is and will have similar dynamics, because it has no continuous symmetry.
Your question doesn't say anything about the presence or otherwise of a gravitational or other potential field which I have assumed exists. If there isn't such a field and no losses then of course your objects will roll on forever.
The speed with which the objects move is of course dependent on the impulse applied. However with a potential field even in a frictionless setting all n-gons will loose energy and come to rest, because of the losses in PE. 855