There is very little evidence for the colors of Roman equipment in any period, especially the Republic. I think the most distinctive color scheme in modern representations is the red scutum shield, and that association between red and the shield probably derives from the Dura-Europos excavations. As far as I know, the only surviving scutum is from there, and it is red. This one is from the 3rd century CE, however, and many think it does not accurately represent regular-use equipment (it is rather finely decorated, and is a little old-fashioned in its shape for the 3rd century). There are other shield fragments from the same site, and they also have a red background. The Palestrina mosaic shows a couple of shields, but both are more yellow than red. It isn't even clear that these guys are supposed to represent proper Roman legionnaires. A late literary source (Vegetius) mentions that different cohorts within the legion used different paint on the shield to identify themselves, but there is no artistic representation of this preserved as far as I know.

A whole other issue is the color of the average soldier's tunic. This debate is centuries old, and the consensus used to be that the soldiers wore red tunics. The assumption then was that a proper "uniform" has matching elements, so if red tunic, red shield and red crest. From there, we get some famous artistic impressions (like those of Peter Connoly) and the conception just sticks.