Yes, you can! Something like this should do the trick:
var cubemeshinstance=MeshInstance.new()
var cubemesh=load('res://cube.msh')
cubemeshinstance.set_mesh(cubemesh)
add_child(cubemeshinstance)
where cube.msh is whatever mesh the user wants to import.
If you want to give it a collider, do the previous from a RigidBody, then do this:
var collshape=BoxShape.new()
collshape.set_extents(Vector3(1,1,1))
add_shape(collshape)
For your application, it sounds like you want to use ConvexPolygonShape or ConcavePolygonShape rather than BoxShape though.