Is it compulsory that every infinite set be non regular?
You must have misunderstood the comment, since the set $\Sigma^*$ is infinite and regular. The correct notion of "infinite" is given by the Myhill–Nerode theorem. Every language is associated with a set of states, and a language is regular iff the set of states is finite.

A state in a language $L$ is a set of strings $S$ that are interchangeable with respect to $L$ in the sense that for all words $x$, either all words of the form $sx$ (where $s \in S$) are in $L$, or all are not in $L$. In the example $L = \{0^n 1^n : n \geq 0\}$, the states are $\{ 0^{m+n} 1^n : n \geq 0  \}$ for each $m \geq 0$, all other words forming another state. You can see that there are infinitely many states, and so the language is not regular.

Every regular language has a DFA whose states correspond to the states described above. This DFA is known as the minimal DFA. A "minimal DFA" for $\{ 0^n 1^n : n \geq 0 \}$ would need to have infinitely many states, since the DFA would have to remember how many zeroes it has seen so far, or more generally how many ones it expects at this point (if the next symbol is a one). No finite automaton can remember that much information, so the language isn't regular.