Would I have all of the rights in Puerto Rico as I would home in Vermont?
Consitutional rights? It’s a commonwealth under the United States, and the U.S. Constitution is in effect. You have all the same rights as the U.S. constitution guarantees you in Vermont. State-derived rights and common-law rights? I’m not sure, as Puerto Rico uses a variation on the civil law, IIRC. JRDelirious, a Puertoriqueño, can speak to that aspect.
You are right that PR is a commonwealth; it is an unincorporated territory which means only parts of the U.S. Constitution are applicable (see this from the Office of Insular Affairs). The only current incorporated territory is the Palmyra Atoll.
Thus, not all of the U.S. Constitution applies. Even if it were an incorporated territory, Vermonters have certain rights that territorial residents would not (representation in Congress, for instance). And of course, the law of Vermont may give rights to Vermonters within the ambit of state power that other states do not give their citizens (such as same-sex civil unions in addition to opposite marriage).
The laws of Vermont and Puerto Rico differ, so any rights that you have under those laws (as distinct from those given the US Constitution) may vary. For example, in Vermont you currently have a right to enter into a civil union with a partner of the same sex, and later this year you will have the right to marry him or her. You would not have those rights in Puerto Rico.
Well, vis-a-vis Vermont, the biggest difference is you won’t have the right to to marry someone of your own sex any time soon. 
Nor to just have as many guns as you like.
Generally, except for some political rights and eligibility requirements/limits for many entitlements, you would be hard pressed to tell while just carrying on a normal daily life that you’re not just in a less laid-back state instead, IRT legal and civil rights specifically. Under rulings and statutes currently in effect (some general, some PR-specific), in everyday practice a Vermonter layman seeking to exercise his or her rights in PR probably couldn’t tell unless advised by his lawyer that anything different is going on than would have in, say, in Louisiana. I sure didn’t feel any less free (in the legal sense) when I moved back from Maryland.
(BTW, “unincorporated territory” is something legislated-from-the-bench by SCotUS in the early 20th Century – try and find anything in the US Code that properly defines what is and isn’t a UI) .
The very, very broad-brush summary is that US-Constitutional Rights that are “fundamental” and accrue to the US citizen (or person under US jurisdiction) as an individual under all circumstances (that’s an important part), apply the same. The US Court for the District of San Juan figures out if your case is one of those or one that is dependent of circumstance.
As to the local level, if a protection has been ruled to the state level under the 14th Amendment, you can assume it’s good in PR, *subject to the specific local legislation to regulate it. *
Rights emanating from Common Law OTOH are another story at the local level: they are there if included as part of a specific constitutional or statutory or jurisprudential provision, otherwise this is a Civil Law jurisdiction. In Commonwealth Court criminal cases are handled US-style, but civil cases are handled under Civil Law.
So to the normal man-on-the-street it means free speech, religion, press, due process, private property, habeas corpus, etc. is good (or as good as in most of the US; after Kelo v. New London, watch out for your house). Guns for private citizens are subject to very strict gun control law. Protected categories for anti-discrimination purposes are essentially the same for the Commonwealth courts as at the Federal level. Federal labor/safety law applies, but it is not a “right to work” jurisdiction. If you actually change your legal domicile to PR you will **not **have the right to vote for US President, Senator or full-rank Congressman, and certain entitlements such as SSI and MediCare will be limited or your elegibility threshold changed. Strictly speaking no Federal Income Tax for individuals whose income is exclusively locally-earned, **but **the Commonwealth Income Tax is all that and more. (Quite more) Etc.
I could be here all night on it, but that’s the gist.
Right. The Constitutional coverage is spelled out in a group of cases called the Insular Cases: Insular Cases - Wikipedia
Which many feel were a travesty, but that’s another half hour. In any case, later Statute recognizing the application of a right in the Territories, and/or any jurisprudence finding that a right is fundamental to the individual citizen and any abridgement subject to strict scrutiny, have whittled away at many of the specific effects of the earlier cases, without overturning the basic idea that not every last part of the Constitution is in force ex proprio vigore in every Territory. By now, as mentioned, it’s mostly lawyers that really notice.
Let me be specific. If I’m in an airport in Puerto Rico and an airport policewoman comes up to me and asks to see my ticket, do I have the right to reply “fish cake” to her (or to simply remain silent)?
Constitutional protections such as the Fourth and Fifth amendments apply, if that’s what you mean.
You have as much right to engage in this conversation with a Law Enforcement Officer at SJU as you do at LGA, LAX or ORD. And the rest of your day will be just as pleasant.
At the actual checkpoint, being a US territory where aviation ops are regulated by the FAA and security of interstate travel by Homeland Sec, the security checkpoint in PR is staffed by the same TSA as in the states (and the ocassional ICE guys looking for illegal aliens or contraband). Now, it may be true that in that case, at the checkpoint “fish cake” would be the reply the TSA person gives to YOU as to what’s the matter (or makes about as much sense as whatever she actually tells you is the matter), before sending you to the rubber glove room, but again that’s about as likely at DFW, ATL or DCA as it is at SJU.
No, this is past the check point, near the gate where the airplane will take off in a half hour or so. If I talk nonsense to this officer, where ever in the US I am, can they stop me boarding the airplane?
If you mouth off to a police officer in an airport, not being able to board the plane will be the least of your worries.
Is that a yes or a no?
Are you asking from the standpoint of what your consitutional rights are? Or are you asking with respect to what the practical aspects are?
And with regard to either, how does that fit into this thread? We understood you to be asking if your rights are any different in Puerto Rico than they would be in Kansas (for example). That question has been answered. If you just want to start a debate on, or get into a discussion of the ins and outs of fourth and fifth amendment protections in general, perhaps another thread would be a good idea. 
Yes. And that is true in America (North, south, middle, etc), Europe, Asia, Australia, and other parts of the planet. I am surprised anyone would believe anything else.
So you are wondering whether you have broader rights in PR than you do in Vermont?
Yes you have the same rights to get in trouble as you would in any other airport.
All the federal laws are the same and all the TLAs (Three Letter Agencies) are present doing the same things they do in the 50.
So yes, you could tell her “fish cakes” (croquetas de pescado, if you think something is getting lost in translation) and yes, you are in your right to not answer their questions. Just as in the mainland, things might go sour really quick, no matter how right you are. Just as in the mainland, they will apologize and let you go once your airplane is in the air.
And they’ll probably give you a complimentary pre-moistened towelette. 
I am a disabled individual with Multiple Sclerosis and it is easier for me to lean against a wall than to sit in a seat, which is what the case was in the San Juan airport about a year ago, when a police officer approached me and asked to see my ticket, for no other reason that I was leaning against a wall rather than sitting in my seat. This pissed me off a bit, but not knowing what my rights were, I obeyed her and showed her my ticket. What would have happened if I decided to f@#k with her and responded to her in nonsense or told her that I wished to consult and attorney before talking to her? Could she have arrested me, or stopped me from boarding the plane?
No one has a constitutional right to board an airplane.