I imagine the people of Britannia were divided on the issue, but I'd love to know more!
Ah yes, the sorry story of the 'Exitus Britannia' (ExBrit, in Classics jargon) - truly a tale of woe and sheer incompetence that has rarely been matched in the centuries since.
The question of Britannia's leaving the Roman Union emerged in the Fourth Century AD. Although discontent had long been brewing at the meddling of unelected Roman bureaucrats who tried to dictate minutiae such as how round an olive needed to be, the immediate catalyst was the lack of control of the border. Specifically, the northern border: Hadrian's Wall had ceased to function as an effective border against the unruly Pictish neighbours, whose predilictions for the odd passtime of 'fitba!' or the strange ceremonial beverage imbibed during the 'Fast of Bucks' made them seem alien to their civilised southern neighbours. As Britons took up the cry of 'Secure our Borders!', political pressure mounted for an official withdrawal from the Roman Union.
Part of the issue was the unwillingness of the Roman Union to create a true 'European Army', that could serve to protect the borders of constituent states. The irony, of course, was that Rome was not actually in charge of Britannia's borders by this point - the regional government in Londinium had been running Hadrian's Wall for decades at that point, but had cocked it up so badly they were happy to let their citizens continue to believe that Rome was to blame for the unwanted migration. Indeed, ExBrit did indeed suffice to discourage migration from within the Roman Empire - but did little to stop the migration of differently coloured (ie blue) people that Britons actually cared most about. Another element that was held against the Roman Union was the lack of expansion eastwards, particularly as the rich, prosperous Eastern Roman Empire - based in Turkey - was not part of the Union. This greatly upset the inhabitants of Britannia, who saw the inclusion of the inhabitants of Turkey as a key selling point of the Roman Union. Lastly, Britannic fishermen seemed annoyed that the inhabitants of a small, unremarkable village in Armorica got to catch more fish than they did - it's not clear why this was the case, though some evidence suggests that these Armoricans were unusually strong.
Even with these differences, ExBrit was by no means assured, and to understand why it happened we need to look to political personalities of the day. Power initially resided with a Roman patrician by name of Cameronus, who was greatly troubled by the divisions among his followers, and the potential for a splinter cult of ardent Romoskeptics led by Nigelus to undermine their power base. He devised a plan: a rare, sacred ritual, involving auguries taken from the internal organs of a sheep and an eel. The results of this consultation would, Cameronus hoped, lay the matter to rest, and keep Nigelus quiet a little longer. Yet Cameronus miscalculated - he had assumed that the auguries would turn out the way he wanted (a similar calculation had worked out in his favour just two years prior!), and he neglected to beseech the gods particularly convincingly to show his plan their favour. Lo and behold, the auguries returned - the omens were divided, but showed a clear path for leaving the Roman Union. The Will of the Sheep-Eel had made itself known.
The omens said little, however, about what this should mean in practice. Cameronus himself resigned in disgrace, the very eventuality he had plotted to avoid. A power struggle broke out: Borus, the court jester, at first seemed likely to triumph, but was stabbed in the back by his erstewhile friend, Brutus Govus. Leo Vulpes, the eternally disgraced, was swiftly eliminated. In the end, Maius Roboticus emerged from the contest as the compromise choice. Maius had supported Cameronus, but indicated a willingness to follow through with ExBrit.
Maius, however, proved a poor choice. No plan for ExBrit could be agreed with the likes of Borus, Ruddus or Moggus, all of whom had very different ideas of what should happen next, and complained bitterly when they thought Maius even hinted at favouring other ideas. Negotiations with an increasingly perplexed Roman Union continued, with few points clarified about what the future relationship might look like. Within the normal course of Britannic politics, another faction would have stepped in to take over, but the most powerful alternative figure - Corbynius - had seemingly little interest in doing so. Maius, when confronted by angry citizens about the chaos, merely repeated ad nauseam that it was 'The Will of the Sheep-Eel', though she did not take the perhaps sensible approach of trying the augury again to make absolutely sure that this was what the gods actually wanted.
The chronicles stop suddenly in 419. We cannot know, but can perhaps assume that Britannic society quietly imploded, to the presumed relief of the rest of the Roman Union who were getting a bit sick of the whole thing. However, they only had a few years to gloat before Putilla the Hun invaded and destroyed them all anyway.
Sources
I have relied chiefly here on the excellent Terrence Proudfish, ExBrit: A Disaster Surely Never to be Repeated, Right? (Edinburgh, 2014). His approach might be balanced with the more critical M. Python, What Have the Romans Ever Done For Us? (Jerusalem, 32).
EDIT: ExBrit is in fact satire, unlike its anagram (somehow) - April 1, 2019.