How do I put a block/union into a player upon them entering the game?

I assume you mean putting a block in the character. This is quite easily done using the PlayerAdded event along with the CharacterAdded event:

local part = game.ServerStorage.Part

game.Players.PlayerAdded:connect(function(player)
	player.CharacterAdded:connect(function(character)
		part:Clone().Parent = character
	end)
end)

This will add the part every time the player re-spawns. If you want it to only add it once when the player enters, you could do this:

local part = game.ServerStorage.Part

game.Players.PlayerAdded:connect(function(player)
	player:WaitForCharacter()
	part:Clone().Parent = player.Character
end)