As Americans, do we have the right and/or duty to prevent an individual from taking the Oath of Office to become the president if they strongly oppose that individual?
The legal right? No.
The moral right and duty? That depends on your personal morals and ethics.
I urge you to remember though, half of your countrymen think that the candidate that won is fit to hold office, and democratically voted for them, twice (nomination and election).
Many of them consider your candidate to be unfit to hold office.
Consider if you want to live in a society where it is OK to overthrow the results of a democratic election.
Yes we do. It's called being civically responsible by being informed, engaged and participating by speaking about the issues, and, most importantly, voting.
Our usual voting rate is around 50% for presidential races, and in this election I heard we had 18 million less votes cast than in the previous cycle. NOTE: Brythan pointed out in the comments that the 18 million number is completely wrong. I'm leaving it there so as not to "whitewash" my original answer, but it is wrong.
So, we have that duty, and we blew it, collectively, if you feel that the outcome is one where an unfit person is going to take office. Too late for buyers remorse after the fact. We had about 15 months to send any unfit candidates packing. That should be plenty of time to figure it out.
The election is over and done with. As Citizens, our job is finished. We've cast our votes and played our part. Legally, we don't have a Do-over button we can smack if we didn't like the result.
As for Morally, it depends. If you believe Trump is Nu-Hitler, then you're morally required to do something. However, I personally believe that representing him that way is overblown fear-mongering, and that from a moral standpoint we need to respect the results of the election.
You do not have the right to overturn a fair and accurate election. You do not have the right to determine fitness. The vote that just occurred did that.
You may want to peruse the laws and penalties against public insurrection: What are the US federal crimes and penalties against public insurrection?
I thought the advocating the overthrow of government particular pertinent in this regard.
Whoever knowingly or willfully advocates, abets, advises, or teaches the duty, necessity, desirability, or propriety of overthrowing or destroying the government of the United States or the government of any State, Territory, District or Possession thereof, or the government of any political subdivision therein, by force or violence, or by the assassination of any officer of any such government; or
Whoever, with intent to cause the overthrow or destruction of any such government, prints, publishes, edits, issues, circulates, sells, distributes, or publicly displays any written or printed matter advocating, advising, or teaching the duty, necessity, desirability, or propriety of overthrowing or destroying any government in the United States by force or violence, or attempts to do so; or
Whoever organizes or helps or attempts to organize any society, group, or assembly of persons who teach, advocate, or encourage the overthrow or destruction of any such government by force or violence; or becomes or is a member of, or affiliates with, any such society, group, or assembly of persons, knowing the purposes thereof—
Shall be fined under this title or imprisoned not more than twenty years, or both, and shall be ineligible for employment by the United States or any department or agency thereof, for the five years next following his conviction.
If two or more persons conspire to commit any offense named in this section, each shall be fined under this title or imprisoned not more than twenty years, or both, and shall be ineligible for employment by the United States or any department or agency thereof, for the five years next following his conviction.
As used in this section, the terms “organizes” and “organize”, with respect to any society, group, or assembly of persons, include the recruiting of new members, the forming of new units, and the regrouping or expansion of existing clubs, classes, and other units of such society, group, or assembly of persons. (June 25, 1948, ch. 645, 62 Stat. 808; July 24, 1956, ch. 678, § 2, 70 Stat. 623; Pub. L. 87–486, June 19, 1962, 76 Stat. 103; Pub. L. 103–322, title XXXIII, § 330016(1)(N), Sept. 13, 1994, 108 Stat. 2148.)
Wikipedia has an interesting article on the Right of Revolution. That right continues to be enshrined in several state constitutions. It's not clear whether it's an individual right (as discussed here) or a collective right (a more widespread uprising)
You have the right to vote your conscience and judgment. You have the right to voice your opinions if you don't like the outcome. However, in a democracy, you don't have the right to disrupt the democratic process if you don't like the outcome. That would be imposing your views upon others, while disrupting due process.
In the US, there are other ways to protest an election. Consider what happened after 2008, when the democratic party swept the presidency and both houses of congress. In 2010, voters went to the polls and gave the House of Representatives, which is fully re-elected every two years, to the republican party. Hint: There is another midterm election coming up in November, 2018.
As for someone being 'unfit for the office'... in the 2016 US presidential election, it is arguable that neither of the prime candidates was fit to hold the office. US voters should be asking themselves how both major political parties could fail so badly, at the same time.
Imagine someone in 2008 deciding that they would disrupt the Oath of Office, because they didn't like the outcome. That might help you get a clearer perspective on what you are suggesting.