Compressive Sensing with Square Measurement Matrix
It's important to clearly distinguish the terms Compressive Sensing (CS) and Sparse Signal Recovery (SSR). CS is about taking fewer measurements than what classical criteria such as Nyquist would dictate us. In the discrete setting this means taking $M&lt;N$ measurements of an $N$-sparse vector.

SSR is about how to recover our $N$-sparse vector from your measurements. 

That said, this should answer your question: If you apply CS, then you need SSR to recover your signal, however, the converse is not true - you can apply SSR even if you didn't do any subsampling. For SSR, your measurement matrix can be flat or square, it could even be tall.

Mathematically speaking, for SSR you solve a problem of the sort $$\min_x \|y - A x \| + \lambda h(x),$$ where $h(x)$ is a regularizer that encourages sparse solutions, i.e., an $\ell_0$-"norm" or its relaxation, the $\ell_1$-norm. In CS, such problems are solved since $A$ is flat and hence we must regularize to find our sparse solution out of the infinitely many possible solutions. The same problem can be used in a non-CS setup where $A$ is square or even tall, e..g, to find an approximate solution that shows a desired level of sparsity.