I apologize if this is a repost, but I couldn't find the question in case it does actually exist here.
I tried and it seems to me that we cannot draw a square with its diagonals without lifting the pen off the paper.
How do you mathematically prove or disprove this claim?
What you are looking for is an Eulerian path (that is to say, a path through a graph that visits every single edge) in the complete graph $K_4$ (the graph with four vertices, in which every vertex is connected).
It is a known property of Eulerian paths that one exists only if there are either exactly zero or exactly two vertices with odd degree (the degree of a vertex is the number of edges it has).
In the graph $K_4$, all four vertices have degree three. Three is an odd number, and thus there are four vertices of odd degree. Thereby, we know that there is no Eulerian path through this particular graph.