Let me start by saying that I'm not a Jewish historian or a Biblical historian and that for a full answer you should wait for someone else, like /u/gingerkid1234. However, I can help a little bit.

First off, there's no reason to believe that these troops weren't actual legionaries. Pilate may have been a member of the equites, but that doesn't mean that he couldn't command troops. After all, both Pompey and Crassus were also equites and they most definitely commanded troops. Traditionally equites held the command of the fleets (at least after they were standardized under Augustus) and after Augustus formed the urban cohorts the prefect of the city, who commanded them, was always one of the equites. The equites were actually the main reason for the rise of the military class in the later empire, because distinguished officers were usually elevated to the class of equites and subsequently became commanders. And under Augustus and his successors the members of the senatorial class who were appointed to military command were rarely patricians or the older families of the equites, but lower-status equites that could be trusted to be loyal with fewer political ambitions. So Pilate was very much capable of commanding legionaries, and there's no reason to think that he didn't. Roman governors were given a detachment of legionaries for security and to maintain order.

That's as far as my expertise takes me, but I hope it's a starting point. I apologize for not actually answering the question, but I hope I've been of some help.

Bond (1998) has some good info on this. She writes that "Since Jews were exempt from military service on religious grounds, the Judaean auxiliaries were drawn from pagan provincials only" (citing a decree of Julius Caesar, recorded in Josephus, Ant. 14.204, 13.251-2)--and these were "drawn predominately from Caesarea and Sebaste."

These auxiliary troops were stationed throughout the province. The majority would remain at the governor's military and administrative headquarters in Caesarea; smaller garrisons were scattered throughout the Judaean fortresses. Although not a strategically important site, Jerusalem's position as the centre of the Jewish cult meant that it too needed to be guarded by a permanent garrison. One cohort was permanently posted in the Antonia fortress; it was commanded by a tribune and may have been partially mounted. These troops carried out police duties in both the Temple Mount and the town itself. Another camp was located in the western part of the town at Herod's palace; numbers here would be increased during festivals by the presence of the governor and a number of Caesarean troops. (Bond, 13)

And auxiliary forces definitely had centurions. “It would become common in the early empire . . . for ex-centurions from the legions, and then young men of equestrian status (the lower nobility), to be offered a series of commands of auxiliary cohorts and cavalry wings in the provinces" (Mason 2008:36). (Kyrychenko [2013] notes that “The auxiliary centurions could be promoted to the post from the ranks of the auxiliary soldiers, from the ranks of the legionary soldiers, or enter the service in that position through direct appointment in recognition of their higher civilian status, such as a magistrate of a provincial town or a member of an aristocratic family.")

Pilate certainly had an army.

Josephus mentions it, for example, "But now Pilate, the procurator of Judea, removed the army from Cesarea to Jerusalem, to take their winter quarters there, in order to abolish the Jewish laws. So he introduced Caesar's effigies, which were upon the ensigns, and brought them into the city;"

This army had ensigns which displayed effigies of Roman emperors (displeasing the Jews in Jerusalem).

Whether this army was composed of legionaries or other forces, Josephus does not mention.