Creating a conditionalPanel and selectInput for each country|RMU separately will be very tedious and (coding) error prone. What you are looking for is a dynamic UI where the choices in a selectInput depend on previous choices.
I haven't tested this because I don't have your data but the following should get you most of the way there. Put the two outputs below in server.R. Then put the uiOutputs in ui.R (note: add comma's as needed). Before even doing that however, make sure to read the Shiny documentation on dynamic ui linked above.
Put in server.R
output$countrySelect <- renderUI({
countryChoices <- subset(NestingArea2, Sp==input$species)$Country)
selectInput("country", "Country:", choices=countryChoices)
})
output$rmuSelect <- renderUI({
rmuChoices <- subset(NestingArea2, Sp==input$species, Country==input$country)$RMU
selectInput("rmu", "RMU:", choices=rmuChoices)
})
Put in ui.R
uiOutput('countrySelect'),
uiOutput('rmuSelect')