The obvious answer is that we do not know, the only texts we have from soldiers are the administrative chits and personal messages that survive in places like the deserts of Egypt and the anoxic muck of northern England, the formulaic inscriptions like tomb stones and discharge sheets, and incidental inscriptions like curse tablets (to be honest I am not certain about the last one). These are a fascinating source of information and provide a wealth of rich detail but they are focused on more mundane matters. We have messages from the soldiers of an Egyptian outpost clubbing together to rent the services of a prostitute, and soldiers trying to get out of duty rotations, but in much the same way that you probably don't write much about your thoughts on civil war in your daily life, neither did those Roman soldiers.
I am being very specific here because the Roman military did indeed form its own distinctive subculture, or I should say subcultures, much as members of modern militaries do. They even saw themselves as distinct and took pride in that--one famous story has Julius Caesar chastising his mutinous soldiers by addressing them as "civilians" ("Quirites"). So while we can easily say there are numerous examples of how Roman society writ large frowned on civil wars, as the other comment noted and as can be seen in other ways such as the effective ban of celebrating victory over other Romans that lasted through numerous wars until Constantine, it is worth pointing out that this is not the voice of the soldiers themselves.
The closest thing we have is a passage from Tacitus Histories (3.25, translation off Perseus) regarding the Battle of Cremona in the civil wars of the Year of Four Emperors:
The conquerors, in the eagerness of pursuit, dispersed themselves over the entire line of road. The slaughter that followed was made particularly memorable through the murder of a father by his son. I will record the incident with the names, on the authority of Vipstanus Messalla. Julius Mansuetus, a Spaniard, enlisting in the legion Rapax, had left at home a son of tender age. The lad grew up to manhood, and was enrolled by Galba in the 7th legion. Now chancing to meet his father, he brought him to the ground with a wound, and, as he rifled his dying foe, recognized him, and was himself recognized. Clasping the expiring man in his arms, in piteous accents he implored the spirit of his father to be propitious to him, and not to turn from him with loathing as from a parricide. "This guilt," he said, "is shared by all; how small a part of a civil war is a single soldier!" With these words he raised the body, opened a grave, and discharged the last duties for his father. This was noticed by those who were on the spot, then by many others; astonishment and indignation ran through the whole army, and they cursed this most horrible war. Yet as eagerly as ever they stripped the bodies of slaughtered kinsfolk, connexions, and brothers. They talk of an impious act having been done, and they do it themselves.
Vipstanus Messala was a generation older than Tacitus but the overlapped in age so this could either be a personal account or a later writing--or, because he was twenty years dead by the time the Histories were written, Tacitus could have made it up. But assuming he did not, as Messala was a commander in that war this could either be a real, literal event that happened or a camp legend that grew among the soldiers. Either way, it points towards the soldiers certainly having a different attitude towards killings in civil wars than in foreign wars. I would even go so far as to say that parallels to this story in other wars indicate a certain universality of this experience, of the literal kinslaying standing in for the metaphorical one.
Of course there is a nuance here: in 69 CE the last time Romans killed Romans in civil war was well out of living memory, Augustus' civil war was concluded more than 80 years ago in 27 BCE (the next full revolt would be even further off, in 195 CE, although there was a bloodlessly suppressed coup in 42 CE). The soldiers at Actium in 31 BCE, on the other hand, would have known little but civil war, as Caesar had crossed the Rubicon in 49 BCE, a mere 18 years earlier. It is not hard to imagine them having a different attitude towards civil war.
The Roman military is unusually blessed with rich source materials, but unfortunately is still subject to the same loss as elsewhere in the ancient world, and so we can't answer your question for certain. That said, I suspect they did not much like it.
Not a direct answer, but /u/sapere_avde 's comment adresses guilt, shame and trauma in Ancient soldiers
Thankfully, we have some commentary on this phenomena from the Romans themselves. A particularly poignant passage is from the Histories, by Cornelius Tacitus. Tacitus is one of our finest sources for the civil wars that occurred during the Year of the Four Emperors (69 CE) -- he was a contemporary of these events, and so can offer an unusually timely and relevant perspective on Roman attitudes during such conflicts. Tacitus writes:
An event which made the slaughter more dreadful was the death of a father at the hands of his son. I record the incident and the names on the authority of Vipstanus Messalla. A recruit to the Hurricane Legion, one Julius Mansuetus from Spain, had left a young lad at home. Soon after, the boy came of age, and having been called up by Galba for service in the Seventh, chanced to encounter his father in this battle and wounded him seriously. As he was searching the prostrate and semi-conscious figure, father and son recognized each other. Embracing the dying man, the son prayed in words choked by sobs that his father's spirit would be appeased and not bear him ill-will as a parricide: the act was not a personal one, and one single soldier was merely an infinitesimal fraction of the forces engaged in the civil war. With these words, he took up the body, dug a grave, and discharged the last duty to his father. Some nearby troops noticed this, then more and more; and so throughout the lines ran a current of wonder and complaint, and men cursed this cruelest of all wars. However, this did not stop them killing and robbing relatives, kinsmen and brothers: they said to each other that a crime had been done - and in the same breath did it themselves.
So we can see that, at least for educated Romans contemplating these internal conflicts, there was a distinct sense that such war was crueler, more anathema, more likely to lead to acts that offended traditional Roman morality.
Which isn't to say that the soldiers themselves had a massive reluctance to fight, or didn't view this conflict as being at least reasonably similar to one against a non-Roman foe. Let's look at the ostensible words of a Flavian officer, Antonius Primus, again related through Tacitus, and again related to the civil wars of 69 CE:
Antonius and his men could now recognize each other. So he seized the chance of spurring them on, some by taunts and appeals to their pride, many by praise and encouragement, all by hope and promises. [...] These were the very battlefields which offered them a chance to wash away the stain of past humiliation and regain their credit in men's eyes. Then, turning to the troops from Moesia, he called on them as the leaders and authors of the campaign: they had challenged the Vitellians by threats and words, but this meant nothing if they were not going to endure their deeds and looks. Such were his arguments to the successive contingents as he reached them; but he spoke at greater length to the soldiers of the Third, reminding them of their early and recent history, how under Mark Antony they had routed the Parthians, under Corbulo the Armenians, and, in the immediate past, the Sarmatians. Then he spoke with greater sharpness to the praetorian guards.
'As for you,' he said, 'you are finished as soldiers unless you beat the enemy. What other emperor and what other camp is there to which you can transfer? There, among the foe, are the standards and equipment which are really yours, and for the beaten the sentence is death**. Dishonour you have drunk to the dregs.'**
Everywhere there were cries of enthusiasm, and as the sun rose, the Third greeted it with cheers in accordance with Syrian custom.
Here, Antonius's men are marching to do battle against other Romans, and we can notice a few things:
So perhaps a legionary would not be particularly loath to engage his fellow countrymen, nor would he see such a conflict as entirely different from one fought against a foreign foe. However, note that Antonius also appeals to the fact that victory is, for the men, existentially required: They are in rebellion against other (legally sanctioned) forces of the empire, and perhaps if they fail in their attempt they will have no chance to be reintegrated into the fabric of martial society. There is a sense, then, that the legionaries are quite aware of the fact that their war is different, its consequences are different, and their enemy is, indeed, the people of Rome.
Let's also reflect on the suicide of Otho, one of the Imperial claimants during these wars. Again, from Tacitus, as he sets the scene:
So eager were Otho's men that they did not wait for the emperor to make any announcement. They told him to be of good cheer, pointing out that his new armies were still intact and that they themselves were prepared to do or die. [...] No one therefore doubts that there might well have been a resumption of desperate and costly fighting, hazardous to victors and vanquished alike.
But Otho, seeing the potential loss of Roman life that looms in from of his country if he counties this conflict, evinces the following sentiment:
'This spirit,' he said*,* 'this courage of yours, must not be exposed to further danger. That, I consider, would be too high a price to pay for my life. [...] We have sized each other up, fortune and I. [...] Civil war began with Vitellius, and with him lies the responsibility for our embarking on an armed struggle for supremacy. I too can set an example by preventing its repetition. Let this be the act by which posterity judges Otho. Vitellius shall live to have the society of his brother, wife and children: I require neither vengeance nor consolation. It may well be that others have held the principate longer, but I shall make sure that no one quits it more courageously. It is not for me to allow all these young Romans, all these fine armies, to be trampled underfoot a second time, to their country's loss. Let your devotion accompany me, just as if you had in fact died for my sake — but live on after me. I must not impede your chances of survival, nor you my resolution. To waste further words on death smacks of cowardice. Here is your best proof that my decision is irrevocable: I complain of no one. Denouncing gods or men is a task for one who is in love with life.'
Then, says Tacitus, "at dawn he fell upon his dagger."
While these speeches are quite likely Tacitus's inventions, rather than verbatim reports of what someone like Otho actually said, we can read within them a commentary on traditional Roman values at the time. And we can see, in Otho's self-sacrificing suicide, the fact that Romans did indeed understand that a civil war was fundamentally more costly, more anathema, more terrible for Rome.
I'm not entirely certain whether Romans felt bad about killing other Romans during the Civil Wars, but I do know that Roman society was heavily divided along political lines during this period to the point that people divided themselves into certain groups based on which politician they supported. During the Roman Civil Wars people and soldiers often were more loyal to a general or politician rather than the Roman government itself, and thus I don't believe many Romans would have felt too bad fighting for their preferred politician against fellow Romans backing a rival politician.
I do know though that there is definitely a sense of "countrymen" between Romans as opposed to a sense of "other" regarding the barbarians. The evidence I have for this is the final triumph Julius Caesar held upon his victory in the Second Roman Civil War.
Caesar basically coerced the Senate into granting him four separate triumphs for his four separate campaigns during the Civil War, one for his Gallic campaign, one for his Egyptian campaign, one for his Asia Minor campaign, and one for his North African campaign. The final North African campaign however was different from the rest in that it was waged against fellow Romans, namely the Pompeians Scipio and Cato. The Romans were a little uneasy about celebrating the deaths of fellow Romans but Caesar appeased them by claiming the triumph was mainly about his defeat of the Pompeians' ally the Numidian king Juba.
However at the triumph, which is essentially a massive military parade where a victorious Roman general shows off the prisoners and loot he won in a war, Caesar had two paintings displayed, one depicting the suicide of Scipio, and one depicting the suicide of Cato. The Roman populace was heavily put off by these paintings, as it appeared that they were celebrating the gruesome deaths of fellow Romans. Additionally, they were upset that despite Caesar's claims the triumph was to celebrate his victory over the Numidians he was mainly portraying the deaths of his Roman political rivals. The crowd turned sour and eventually turned violent after Caesar paraded a young child king before the crowd with the alleged intention of having the child publicly executed.
I believe this incident shows there was some sense of camaraderie amongst the Roman populace. It is telling that the Roman people got so upset at Caesar because he seemed to be relishing and praising the gruesome suicides of two prominent Roman politicians. While they may not have had trouble killing Romans who they believed were political rivals, they did not enjoy celebrating and praising the death of fellow Romans as much as they did with the so-called barbarians. I think this shows that yes, there was a concept of "countrymen" amongst the Romans, at least in regards to fighting fellow Romans as opposed to fighting barbarians. The death and defeat of barbarians was something to be publicly celebrated, but the deaths of fellow Romans was not.