What are the proper terms for "square" and "non-square" tensors? Is "square" implied when referring to tensors? To my understanding, mathematically, there's no reason for "non-square" tensors to not exist..
What I mean by square tensor:
Operates on one vector space and its dual space like $V^{*} \times V^{*} \times V^{*} \times \dots \times V \times V \times V \times \dots$
What I mean by non-square tensor:
Operates over different spaces like $U \times V \times W \times \dots$
There isn't any special terminology for your "non-square tensors". They are usually just called "multi-linear maps" while the "squared-tensors" are sometimes called $k$-covariant, $l$-contravariant (the numbers $k,l$ marking how many factors of $V$ and $V^{*}$ appear) tensors on $V$. A uniform notation for both tensors can be
$$ \operatorname{Mult}(V_1,\dots,V_k;W) := \{ T \colon V_1 \times \dots \times V_k \rightarrow W \, | \, T \text{ is multi-linear} \}. $$
Using this notation, your "square" tensors are elements of $\operatorname{Mult}(V^{*},\dots,V^{*},V,\dots,V;\mathbb{F})$ while the non-square tensors are elements of $\operatorname{Mult}(U,V,W,\dots;\mathbb{F})$.