I can only click on one of my gui classes?

Your script is stopping because while true do will begin running when it loops through the first button so next loops aren't ever going to begin, your solution is not use a loop, instead keep a track of what class user is on the server and when he spawns, give him that class.

local rp = game:GetService ("ReplicatedStorage")
local Class = rp:WaitForChild ("Class")
local players = game:GetService("Players")

local SS = game:GetService ("ServerStorage")
local Classes = SS:WaitForChild("Classes")

local playerClasses = {}

local function setClass(Player, className)
    local selection = Classes:FindFirstChild(className)

    if selection then
        local moveset = selection:Clone()
        -- I believe it's better to use WaitForChild on backpack
        -- since it caused weird behavior when not using it for me
        -- but it's up to you
        moveset.Parent = Player:WaitForChild("Backpack")
    end
end

Class.OnServerEvent:Connect(function(Player,className)
    setClass(Player, className)

    -- save last used player class into a table
    playerClasses[Player] = className
end)

players.PlayerAdded:Connect(function(player)
    player.CharacterAdded:Connect(function(character)
        -- get last used player class from the table
        local className = playerClasses[player]

        -- If player does not have any previous class then
        -- just return and don't continue the function
        if not className then
            return
        end

        setClass(player, className)
    end)
end)

-- When player leaves remove his last saved class since
-- it won't be used anymore so there is no point in saving
-- it, removing this will cause memory leaks on big servers
players.PlayerRemoving:Connect(function(player)
    playerClasses[player] = nil
end)

And for the LocalScript:

local Background = script.Parent
local rp = game:GetService("ReplicatedStorage")
local Class = rp:WaitForChild("Class")

for _, button in ipairs(Background:GetChildren()) do
    if button:IsA("ImageButton") then
        button.MouseButton1Click:Connect(function()
            Background.Visible = false
            Class:FireServer(button.Name)
        end)
    end
end

Use ipairs for arrays, it improves performance.

Delete the while true do line in LocalScript. It's unneeded.

local rp = game:GetService ("ReplicatedStorage")
local Class = rp:WaitForChild ("Class")

for _, button in pairs(Background:GetChildren()) do

    if button:IsA("ImageButton") then
        button.MouseButton1Click:Connect(function()
            Background.Visible = false

            Class:FireServer(button.Name)
        end)

        local player = game.Players.LocalPlayer

        local hum = player.Character:WaitForChild("Humanoid")
    end
end