There is some evidence to support it. Red was considered a martial colour (quite literally, Mars was clad in red, and he was as red as was Ares), divine and awe-inspiring.

One of Martials epigrams (14, CXXIX) mentions a soldiers preference for the colour red:

Roma magis fuscis vestitur, Gallia rufis,
Et placet hic pueris militibusque colos.

'Rome rather clads itself in dark colours, Gaul in red, and this colour boys and soldiers also like'.

Then there's Plutarchs 'life of Fabius Maximus':

But Terentius, insisting on his right to command a day in turn, and then encamping over against Hannibal by the river Aufidus and the town called Cannae, at break of day put out the signal for battle, — a scarlet tunic displayed above the general's tent.

Other writers who mention red clothing for Roman legionaries are Isidor of Sevilla (who is often considered not very reliable) and the Historia Augusta, where red is mentioned as the colour that Roman leaders (duces) wore. This doesn't mean that all Roman soldiers wore red tunics all of the time, though this has been argued in the past.

Polybius mentions a purple tunic:

Caesar, therefore, after saying that the expected day had come, on which they would fight against men, and not against want and hunger, quickly ordered the purple tunic to be hung up in front of his tent, that being the Roman signal for battle.

Roman leaders (dicators, imperatores, legates, military tribunes) commonly wore the paludamentum, a quadratic cloak that was fastened with a brooch over one shoulder. This could be crimson, purple or white in colour. Pliny (n.h. XX, 3) mentions crimson as a dye for officers cloaks.

Interestingly, Monica Cyrino in Big Screen Rome makes the claim that the Romans wore Red tunics because they usually, in the Hollywood productions of the 50s and 60s, represented the godless, degenerate antagonists - so the colour red was chosen to emphasize a parallel to the 'godless, degenerate' communists. Kubrick's Romans in Spartacus wear red tunics too, though. Red probably simply looks very striking on the silver screen. They are not always portrayed red; the Asterix comics, for example, picture the legionaries in green tunics, and the officer here is wearing a red cloak.

It's not helpful to approach the topic with the modern notion of a 'uniform'. Though the equipment in later times was not as diverse as in republican times, there were still variations. Even in modern armies, we see soldiers wearing different uniforms, and the famous British 'redcoats' were often found in blue or green. The Roman legionary received a lot of new clothing during the year, as payment accounts tell us, they were probably not all of uniform quality or colour. There is no definite account that tells us 'roman legionaries wore red'. Many may have some of the time, but there's no evidence that all did all of the time. Vegetius (IV, 37) mentions that rowers, sailors and marines/naval soldiers wore blue tunics, for example. Some scholars suggested that Roman soldiers wrote white tunics in battle.

So what does this tell us? If this seems a bit confusing, that's because it is. Like so often with antiquity, there's no clear consensus, not much evidence and even less hard facts - If you want to read more on it, I suggest Graham Sumners Roman Military Dress, which is probably the most exhaustive volume written about this topic so far.