I know that Kepler thought based on comet tail that light might exert pressure (although is the Solar wind not also involved?) but did Maxwell's prediction stem from newer observations or from perhaps his equations?
According to Brian Clegg's biography of Maxwell the idea that light could exert pressure only came to Maxwell's attention shortly after the publication of his Treatise while "pulling at the threads of new ideas" suggested by his theory.
Maxwell gave this hypothesis a theoretical backing by showing that the energy of absorbed light should result in momentum being added to the body absorbing it.
The reason we don't see this happening as a rule, Maxwell suggested, was because the effect was so small. He calculated that the sun, the most powerful light source in our vicinity, would only produce the equivalent pressure of $7$ grams ($0.015$ pounds) across a whole hectare ($2.47$ acres) of area. Maxwell would never see a demonstration of the effect that he had predicted, but $25$ years after his theory was published, the Russian physicist Pyotr Lebedev demonstrated this 'radiation pressure' for the first time.
Clegg also writes that Maxwell, in a letter to the Harvard astronomer George Phillip Bond dated 1863, some ten years before the publication of his Treatise, wrote:
Is there anything about a comet to render its lines of force visible? and not those of a planet which are much stronger? I think that visible lines of gravitational force are extremely improbable, but I never saw anything so like them as some comet tails.
So it is clear that Maxwell was considering the curious properties of comets before he arrived at the idea of light exerting pressure. However, his motivation for the claim that light exerts pressure appears to have been born of purely theoretical considerations.