Change the color of action button in shiny

You can use boostrap colors in the class attribute:

actionButton("run","Run Analysis", class = "btn-warning")

These are basic colors only but really usefull for graphic standards.

Below, I've made your action button look like a submit button (also adding a font-awesome icon):

actionButton("run", "Run Analysis", icon("paper-plane"), 
    style="color: #fff; background-color: #337ab7; border-color: #2e6da4")

As @MLavoie mentioned, you can embed CSS in your shiny app using tags$head. Try this:

library(shiny)

ui <- shinyUI(fluidPage(
  tags$head(
    tags$style(HTML('#run{background-color:orange}'))
  ),
  actionButton("run","Run Analysis")
))
server <- shinyServer(function(input, output) {

})
shinyApp(ui, server)

If you're unfamiliar to CSS, w3schools has really good and easy tutorials.