Can I use modulo $n^2$ arithmetic where $n=p \cdot q$ for ElGamal encryption? In the paper which I was reading, the authors were using ElGamal encryption with modulo $p$, where $p$ was a prime.

Do the commitment properties still hold if we use modulo $n^2$ where $n=p \cdot q$ and $p$ and $q$ are large primes?

I was asking this because a number of other commitment schemes use modulo $n^2$ arithmetic.
To completement the answer of SEJPM, note that ElGamal should not be used directly over the ring $\mathbb{Z}_{n^2}$ (more precisely, over the multiplicative group $\mathbb{Z}_{n^2}^*$), because it is insecure here. To be more precise, it is still one-way, which means that if you get an encryption of a random plaintext, it's hard to recover the plaintext. However, it is not IND-CPA secure anymore, because the IND-CPA security property of ElGamal reduces to the Diffie-Hellman assumption - but the Diffie-Hellman assumption does not hold over $\mathbb{Z}_{n^2}^*$. Intuitively, the reason behind this is that computing the Jacobi symbol is easy, and given $(g^a, g^b)$ where $a,b$ are some exponents, and $g$ is some element of $\mathbb{Z}_{n^2}^*$, it holds that 

JacobiSymbol$(g^{ab}) = $ JacobiSymbol$(g^a)\cdot$JacobiSymbol$(g^b)$

which allows to break the decisional Diffie-Hellman assumption with non-negligible probability (with probability 1/2, this relation will not be satisfied with a random tuple, while it is always satisfied for a Diffie-Hellman tuple).

This can be fixed by working into proper cyclic subgroups of $\mathbb{Z}_{n^2}^*$ where elements have all Jacobi symbol 1 - the DDH assumption is believed to hold over such groups, so ElGamal should be secure. However, note that it will be far less efficient than standard ElGamal over prime order groups, and should only be used if it is part of a larger system where you really need to use the specific structure of $\mathbb{Z}_{n^2}$.