Can a Dem/Rep primary losing candidate run as a third party candidate?
"Once the Primary is over. If a candidate does not win in the party they ran in can he/she switch to lets say a different party like Libertarian or some other open party to run against the Republicans and Democrats in the run off?"
Yes. If a candidate does not win the primary, he/she can run as a third party candidate or write-in in the general election (not runoff). HOWEVER, between ballot access issues (requirements/deadline to get on the ballot in each state) and the Electoral College, the process makes it near impossible to win. Essentially, the game is rigged to Democrat & Republican.
On the other hand, a primary losing candidate may still have strong enough appeal in the public eye. In this case, he/she can run as a third party candidate or write-in, be a spoiler, and change the outcome of an election.
Think Ross Perot and Ralph Nader in recent elections.