Usually both; although the army as a whole might carry a banner representing their employer, individual companies would want to use their own banners, since those would be the ones their men were most familiar with. However, there was no all-encompassing rule; different combinations of personal and political banners could be used, and different captains might even change their standards over the course of campaigns as alliances and allegiances shifted. An example is Giovanni de' Medici: although he was active during the early modern era, he stands out for having painted his banners black after the death of his uncle and employer, Pope Leo X. Giovanni had been in Lombardy as part of a Papal Army commanded by Prospero Colonna: the Papacy had sided with Charles V in a plot to expel the French occupiers from the Duchy of Milan and prop Francesco II Sforza (incidentally Giovanni's maternal cousin). Although Francesco was successfully propped up in Milan, Giovanni refused to return to Rome, and instead his company stayed in Lombardy to help his half-sister, the widowed Countess of San Secondo, assert her right to her late husband's fief against the claims of an ambitious relative.

Even though Giovanni would later return to serve under the Papacy, he would forever thereafter be known as Giovanni Dalle Bande Nere, meaning Giovanni of the Black Banners.

Some employers, notably the Republic of Venice, made a big ceremony of empowering commanders with banners, which in Italy were called "Gonfalone" (or, more accurately, in the plural form: Gonfaloni). This was particularly marked even at the individual company level, because the Venetian army was slightly less dependent on individual captains to raise fighting men: in times of war Venetian cities could be tasked with raising companies who would usually fight under their city banner (sometimes also conducting similar ceremonies to that in Venice) in addition to whatever units were raised by individual condottieri. However, here too there was no all-encompassing rule; although a Captain-General presented with a city's banner might be appointed to oversee that individual city's contribution to the Venetian war effort, that city might also delegate raising companies to individual captains who have their own banners.

At the end of the day, a banner or pennant needs to serve as a rallying point; a reference point for fighting men so they can keep good order. How that reference point was constructed could depend on the needs of the time, the dynamic of the particular army in question, and the whims of individual commanders.

Edit: All this is, of course, with definitional problems regarding mercenaries set aside. I actually wrote a similar answer that goes into more depth here that also provides more examples.

The answer is that they carried both, but is complicated by the issue of how you define a mercenary.

To answer the question directly, though, yes, mercenaries did fight under the banner of their employer, or else wear his badge or his colours, because they were symbols of their employers authority and authorised them to use force. Without some symbol of their employer, a mercenary was not a legitimate power. For that reason, when an unnamed outlaw in the 13th century attempted to outfit his band with coats all of the same colour and cut, the nobility was outraged. Here was a peasant, a criminal, acting as though he had a right to declare war and utilise violence. Similarly, the unfurling of banners by the English nobility without the permission of the King was banned and seen as a sign of treason, and when rebellious towns on the Continent were defeated, they lost their right to have a banner.

As an example, when Bertrand du Guesclin organised the "free companies" which were ravaging France and led them to fight on a "crusade" in Spain, they were all outfitted with a white cross on their clothing. This was not just a symbol of the Faith, it was also the unofficial emblem of the French monarchy, and had been in use for quite some time to distinguish French forces from the English, who wore a red cross.

But, at the same time, within any mercenary company would be knights or, at least in England, squires, who all had their own coats of arms and symbols. Of these, some would be bannerets, knights who were permitted to fly their own banner. These bannerets would frequently be in charge of sections of the force, through both social standing and experience. And if a knight, banneret or higher lord led a mercenary troop, then their banner would become the default symbol of the company as a whole.

The question is, to what extent do you consider a knight a mercenary? Not infrequently they fought in nations other than that of their birth for money, rather than any land they held there, yet that's not always the full picture. Are the household knights of a lord, some of whom might be foreign, mercenaries or not? They have no land and are paid for their service, yet are part of the social fabric, even if foreign. Would the Flemish knights theoretically available to the English kings in the 12th century be mercenaries, through their political uninvolvement, or should they be considered a part of the feudal levies the King had access on account of the alliance between Flanders and England.

Or what of the lords and captains of England who lead contracted companies into France during the Hundred Year's War? They might owe some service and obligation to the Crown, but they fought for money, and so did their contracted soldiers, most of whom were native English. At what point does serving ones king become pure paid mercenary service?

Edit: Stephen Morillo's quadrant system for determining types of service

Follow up question if i am allowed: was there a specific person thatwas tasked to hadle/carry/wave(?) The banner during battles and marches? If so, what would happen when they died, would the banner be passed on to someone specific in the regiment, or just to some bloke in the center or one of the corners?

As a follow-up- Is there any resource that has banners that were used throughout history? That would be a cool book/website