There's no reason for your code to be so complex!

Light Script

First, you use script.Parent.PointLight 11 times. Use a variable, say, light.

You continually have this pattern:

    if X == A then
        Y = A
    elseif X == B then
        Y = B
    elseif X == C then
        Y = C
    ...

Instead, just say before the ifs and only once:

    Y = X

that is,

    ....
    color.BrickColor = screen.BrickColor
    if screen.BrickColor == BrickColor.Blue() then
        ....

A similar thing can be said for enabling / disabling the light. Since only one disables the light, you might as well just default to enabling, and only changed Enabled for Black:

    light.Enabled = true
    if screen.BrickColor == BrickColor.Blue() then
        ....
    elseif screen.BrickColor = BrickColor.Black() then
        light.Enabled = false
    end

At this point, we still have a really large if .... elseif thing going on, which probably isn't what you want.

One option at this point would be to ask BrickColor for the Color3 values; these are different from what you've picked, but depending on why you picked these in particular, that might be okay.

That would make the entire function this:

function onClicked() 
    color.BrickColor = screen.BrickColor
    light.Color = screen.BrickColor.Color
    light.Enabled = true;
    if screen.BrickColor == BrickColor.Black() then
        light.Enabled = false
    end
end

In fact, you could shorten it a little by removing the if:

function onClicked() 
    color.BrickColor = screen.BrickColor
    light.Color = screen.BrickColor.Color
    light.Enabled = ( screen.BrickColor ~= BrickColor.Black() )
end

If you wanted custom colors, you could keep it how it is, or you could use a dictionary.

local color3s = {
    ["Bright blue"] = Color3.new (46/225, 43/225, 149/225),
    ["Dark green"] = Color3.new (34/225, 97/225, 25/225),
    ["Bright red"] = Color3.new (207/225, 11/225, 0/225),
    ["Bright yellow"] = Color3.new (236/225, 189/225, 34/225),
    ["White"] = Color3.new (255/225,255/225, 255/225)
};

function onClicked()
    color.BrickColor = screen.BrickColor
    light.Color = colors3s[ tostring( screen.BrickColor ) ]
    light.Enabled = ( screen.BrickColor ~= BrickColor.Black() )
end

Button Script

Again, we find ourselves repeating a lot. At the least, we could make a variable for screen.BrickColor:

local old, new = screen.BrickColor
if old == BrickColor.Blue() then
    new = BrickColor.Green()
elseif ....
end
screen.BrickColor = new

That's not really the best. What we're doing here is defining some sort of cycle, so we should just store it in a list and use a loop to figure out what's next. (We can also make it briefer by using the string names)

local cycle = {
    BrickColor.Blue(), BrickColor.Green(), BrickColor.Red(),
    BrickColor.Yellow(), BrickColor.White(), BrickColor.Black()}

function one()
    for i = 1, #cycle - 1 do -- Not black...
        if screen.BrickColor == cycle[i] then
            screen.BrickColor = cycle[i + 1] -- Next in cycle
            return -- I'm done; I found the match
        end
    end
    -- since black is last a simple + 1 won't be enough
    screen.BrickColor = cycle[1] -- First in cycle
end

We actually can pack up the last element into there pretty easily; just make a copy. Since we stop at the first one, it's okay to have duplicates:

local cycle = {
    BrickColor.Blue(), BrickColor.Green(), BrickColor.Red(),
    BrickColor.Yellow(), BrickColor.White(), BrickColor.Black()}
table.insert(cycle, cycle[1])

function one()
    for i = 1, #cycle do
        if screen.BrickColor == cycle[i] then
            screen.BrickColor = cycle[i + 1]
            return
        end
    end
end

Beyond This...

I don't see any particular issues with the code.

If there are problems, I'm guessing it has to do with objects not loading in the order that you expect / want them to.

Check your output, print everything that's suspicious.

You haven't said what exactly is wrong. What does act up mean?