According to Catholicism, does reception of the gift of the Holy Spirit protect against demon possession? A Catholic answer to this question regarding instances of animal-like behaviors in spiritual movements in history included reference to the Loudon possessions in 1634 in France in which a group of nuns were possessed:

The nuns claimed the demon Asmodai was sent to commit evil and impudent acts with them. During questioning about the supposed evil spirit thought to be possessing them, the nuns gave several answers as to who caused its presence: a priest, Peter, and Zabulon.
The nuns' increasingly extreme behavior: shouting, swearing, barking, s etc. drew a considerable number of spectators.

A comment questioned the implications for the salvatory state of the nuns:

An interesting thing is that the Scripture strongly indicates that born-again believers (indwelt by the Holy Spirit) cannot be possessed by demonic spirits, so what does that indicate for the Nuns?

The responding comment was that Catholic exorcists and Catholic hagiography indicate differently:

Catholic exorcists would genuinely disagree with that opinion. This is not the place to start an argument, on this subject. Catholic hagiography actually supports it, as happened to even a few canonized saints. The Devil hates the truth!

My question is: Does the Catholic Church actually teach that believers who are indwelt by the 3rd person of the Trinity can still be demon possessed?  Do Catholics have reason to fear that they might get possessed?
According to Catholicism, does reception of the gift of the Holy Spirit protect against demon possession?
Actually Catholics believe anyone could be possessed by evil if the circumstances are there. That includes Catholics and any other Christians.
First of all let us clear up the notion of “born-again” in the mind of Catholic thought. I do this not out of desire to create an argument, but rather to stress how Catholicism sees things, after you quoted my comment here.
The Early Church Fathers understood born-again as having being baptized. This is the way my comment should be interpreted, and was my though when I responded to it. Has the term “born again” always been synonymous with baptism with the Christians of the Early Church? It is basic Catholic theology that when one is baptized, he or she receives the graces of sanctifying grace and becomes an dwelling place for the Holy Spirit. This holds, in Catholic thought, for anyone baptized whether Catholic, Orthodox, Anglican, Protestant or any other validly baptized Christian.
Historically, we know that the Devil has possessed Catholics, even canonized saints have been possessed through no fault of their own.
Most diabolical possessions occur because someone is dealing with the occult, or has been cursed by someone. Possession may occur because of any number of reasons. In fact, one of the few questions an exorcist must ask the Demon is the reason for the possession itself.
As a title of example there is Saint Mary of Jesus Crucified who was possessed by the Devil twice. The last time was for 40 days. In her biography, it read almost like the Book of Job, where the Demon asks Jesus to possess this nun in order to make her fall into sin. It failed! In fact the Devils wanted to leave before their time was up. In the end, St. Mary of Jesus Crucified gained more graces and her love of the Divine Saviour increased.

She continued to experience supernatural events. She fought off a demonic possession for 40 days, received the stigmata, was seen to levitate, had the gift of prophecy and knowledge of consciences, and permitted her guardian angel to speak through her. Helped found the missionary Carmel of Mangalore, India. Returned to France in 1872. Built a Carmelite monastery in Bethlehem in 1875. Supernatural gifts aside, she was known for her devotion to the Holy Spirit, even sending word to Pope Pius IX that the Spirit was not emphasized enough in seminaries. - Saint Mary of Jesus Crucified

Here follows how most causes of demonic possession are achieved (The Source is not the best and this list is not the best in it’s English explication, but for the present moment it will do. It basically is what the famous exorcist Gabriele Amorth states in his books):

Causes of Demonic Possession
According to the Catholic Church, the chief causes of possession are the following:
• making a Pact with the Devil or Demons
• participating in occult or spiritualist rites, including playing with divination devices such as a Ouija™ or doing automatic writing
• offering or dedicating a child to Satan
• being the victim of a witchcraft spell or Curse
Engaging in these activities, as well as leading a deliberately sinful life, give Demons the right and license to take up residence, according to the church. Mental illnesses such as schizophrenia and multiple personality disorder are not considered to be caused by Demonic possession.
The church teaches that God allows possession to happen for a variety of reasons:
• to Demonstrate the truth of the Catholic faith
• to punish sinners
• to confer spiritual benefits through lessons
• to produce teachings for humanity

Any Christian, whether a Catholic or a “Born-Again” Christian who deliberately plays around with the occult, is in serious threat of being diabolically possessed. We are all human and capable of doing very serious offences against God and our neighbour. We are all able to bless those who bless us, but one bad egg in the group could actually curse another.
Therefore, let us live up to St. Paul’s desire and become saints in all our ways.

Throw Out the Evil Person!
5 It is actually reported that there is sexual sin among you. I’m told that a man is sleeping with his father’s wife. Even people who don’t know God don’t let that kind of sin continue. 2 And you are proud! Shouldn’t you be very sad instead? Shouldn’t you have thrown out of your church the man doing this? 3 Even though I am not right there with you, I am with you in spirit. And because I am with you in spirit, I have already judged the man doing this. I have judged him in the name of our Lord Jesus. 4 So when you come together, I will be with you in spirit. The power of our Lord Jesus will also be with you. 5 When you come together like this, hand this man over to Satan. Then the power of sin in his life will be destroyed. His spirit will be saved on the day the Lord returns. - 1 Corinthians 5:1-5

St. Augustin of Hippo warns the faithful to keep a good distance from the Devil.

He (the devil) is tied up like a dog on a chain, and can only bite someone who, deathly sure of himself, goes near him. Wouldn’t you think a man a fool who let hiimself be bitten by a chained up dog?
He can only bite those who willingly let him. It is not by force, but by persuasion, that he harms: he asks for our consent, he does not drag it from us. Augustine on the binding of the devil