The Constitution may not prohibit them from running, but there is nothing that says that they have to be allowed to run on a particular party.
That is true. States, have always set restrictions for other parties that are not Republican or Democrat. Primaries, or getting on any ballot, always have a different standard. Hans is suggesting, that parties not have standards, and that all parties are equal. Perhaps that should be the case, and we have hundreds of candidates for every election.