It'd be possible for one to exist for a time, but a naturally occurring, non-spheroid planet would be incredibly unlikely. More on that at the end.
It's an easy thing to imagine of course, but that's because we think of things like cube shaped rocks that occur naturally and think 'why not?'. The problem is that objects on a planetary scale don't behave the same as boulder or even small moon sized things. Anything above 200-300km radius squishes into a spherical shape.
So, you can have a space potato that's not a sphere, but it's technically not a planet. It could have an atmosphere at any size, though it's less likely to hold on to it if there are other large masses nearby or in high solar winds.
To have something the size of a planet but not a sphere you'd have to find a planet that has had a terrible accident. If you crash a large moon into a planet you might be able to horribly deform the planet and give it some new moons. Assuming the remaining mass is large enough it could still be defined as a planet (massive enough to collapse into a sphere) but non-spherical. The non-spherical shape would be temporary, hundreds of thousands of years perhaps. As it's just been destroyed it's unlikely to have life though.
What if Earth were a cube?
There is an interesting article on that here. Basically the corners and edges of the cube would be like massive mountain ranges, as they would be at an angle with respect to the vector of gravity (except in the center of the faces). I wrote a more detailed answer about such a world here.
The oceans would move to the centers of the cube faces. Basically, the Earth would look very different. Likely still inhabitable, assuming whatever magic formed it into a cube kept the the shape, otherwise we'd probably all be killed by the planet shaking earthquakes.