It can happen, but it's doubtful that they'd become established. There are two factors that may negatively impact them: the constant humidity and heat, which their skin (which is surprisingly thick) and fur aren't quite suited for (they'll grow thicker fur for "winter" regardless of geographic location) and which would also increase the risk of infection and disease, and food availability. Although there is plenty of edible stuff in a highly productive rainforest ecosystem, grey squirrels are instinctually hard-wired to store food, which would be a waste of energy in a place where food is available year-round.
They'd also have to compete for tree cavities that are already in high demand and/or occupied, since leaf nests would decompose much faster in a humid tropical environment. There may also be issues regarding breeding if hormone regulation and estrous in females is triggered by cold winter temperatures.