What was the military background of the Irish company at the Siege of Jadotville?

You should be aware that Ireland was a neutral country during the Second World War, and Ireland had no overseas colonies. As far as I can see, none of the 151 soldiers in "A" Company of the 35th Battalion of the Irish Army had any experience of active service. This was certainly not a "commando company" or other specialist unit.

While most of the men deployed to Jadotville were 'raw recruits', with almost no experience beyond their basic training when they arrived in the Congo, they did have senior officers and NCOs with experience of leading men, and military exercises. These men understood tactics, even if they lacked combat experience.

The men were given further training and drilling after their arrival in country (see below).

Although heavily outnumbered, the men of A Company had taken the opportunity to dig trenches and construct defensive positions. This gave the men under Commandant Patrick “Pat” Quinlan an important advantage.

The attacking Katangan gendarmes and foreign mercenaries would be forced to expose themselves if and when they took the initiative and attacked the Irish positions. A map showing troop movements and positions at Jadotville was produced by one of the Irish soldiers:

Click to enlarge

• Source South Dublin County Council, CC BY-NC-SA 1.0

The article How 155 Irish soldiers became heroes at the Battle of Jadotville by Brendan Farrel, published on Irish Central (the article that I linked to from my earlier comment), explains why the men of A Company were able to perform as well as they did.

Briefly stated, they were brave men, well led, and had constructed a relatively strong defensive position. They were also lucky (explaining why they suffered so few casualties, and no fatalities).

In his 2006 book, The Battle of Jadotville, Michael Whelan states:

Ireland was asked by the United Nations to contribute troops and in reply the first Irish peacekeeping force began to assemble during July 1960. The Curragh Camp, Co Kildare, was used as a training and forming up facility for the troops. Out of 3,000 soldiers who volunteered from around the country, 689 men were selected to form the 32nd Irish Infantry Battalion. This formation was destined to become the first battalion of Irish soldiers to serve outside the state of Ireland since its foundation.

• p22 (my emphasis)

Chapter 4 of Whelan's book (pp 33-54) has a reasonably detailed account of the battle,

"... based on interviews with veterans of the Congo mission, the unit history of “A” Company, 35th Battalion and the testimony of its commander, Commandant P. Quinlan"

which explains how events unfolded, although, as the author notes, the

"... account has been shortened somewhat by selecting the most salient points. It is intended to chronicle the events and hardships endured by this group of Irishmen."

In addition, a series of interviews with survivors from the Irish soldiers that served in the Congo were published in History of War Magazine. You may find these of interest:

Tom Garner: The Real Siege of Jadotville Part I: Teenage Peacekeeper John Gorman Remembers, 17th October 2017

Tom Garner: The Real Siege of Jadotville Part II: Lieutenant Noel Carey Recalls the Pressure of Command, 30th October 2017

Tom Garner: The Real Siege of Jadotville Part III: Veteran Tony Dykes on the Fight for Elisabethville, 29th November 2017

Finally, I found a 2016 post by someone who identified themselves as:

"... a History student from the north of Ireland, studying history at UUC"

Which contains some interesting images and observations relating to the battle. They also make the point that:

For the Irish Defence Force it was to be their first piece of active combat since 1923.

(i.e. since the Irish Civil War)

On Commandant Quinlan, they wrote:

Patrick Quinlan was the commander of A Company of the 35 Battalion. He was as much a scholar as a soldier and read extensively. He was also a 'true believer' in UN peace keeping and volunteered for the Congo mission.

Cmmdt Quinlan was brutally aware of the inexperience of himself and his troops so as soon as he and his men stepped off the plane he began a strict regime of training and drilling which would prove to be useful later at Jadotville. He demanded excellence, especially in marksmanship and riot actions, and finally in sheer physical fitness, often ordering his officers to do laps of the Elizabethville airport.