How can I start and stop a loop using a button?

Use the while loop.

Make sure there is a clickdetector and that its parent is the button.

I made the script for you:

local button = script.Parent
local clickdetector = button.ClickDetector
local toggled = false


clickdetector.MouseClick:connect(function()
    if toggled == false then
        toggled = true
        while toggled == true do
            --your script here
        end
    end
    if toggled == true then
        toggled = false
    end
end)

Use break. break is used for any type of loop, for loops, repeat-until loops, or while loops. break is self-explanatory, it breaks the loop.

local clickDetector = workspace.Part.Clickdetector
while wait() do
    clickDetector.MouseClick:Connect(function()
        break
    end)
end