If all you want to do is see the voltage levels of the signal then your analysis is correct. Also, in the case of a true square wave (rather than a stream of digital pulses) you will be able to determine the fundamental frequency, according to Shannon and Nyquist. However, a square wave has significant frequency content at much higher frequencies...at least to 11 times the fundamental for practical purposes.
Unfortunately, you will be missing a lot of the information that is critical for digital signalling protocols. You won't be able to determine the actual width of a pulse very accurately. It will also be very difficult to observe the relationships between signals, such as clock and data, that are absolutely vital when debugging serial protocols. To be useful for debugging you must sample at a much higher rate, and you should be able to trigger the scope on the edges of one signal while sampling another signal simultaneously.
My opinion is that a good scope would be sampling at about 20 times the data rate.
EDIT: For serial communications the pulse width, in time, for each bit is a very important parameter. If your data rate is 1Mbps then you expect the pulse width to be 1\$\mu\$s, but if you only sample that signal at 5MHz then your pulse width measurements will be \$\pm\$0.1\$\mu\$s which is pretty rough. More important in my mind are setup and hold measurements. You want to know how long the data signal is stable before a clock edge and how long it remains valid after a clock edge. Specifications for setup and hold may be tens of nanoseconds for a 1Mbps data rate, and it will be very difficult to observe these when sampling every 200ns.
For debugging serial communications the instrument you really want is a logic analyzer. These devices can sample data at very high frequencies but they don't try to measure the actual voltage of the signal, only whether it is a valid logic 1 or 0. They also have other capabilities that are made possible by treating the inputs as digital data rather than analog voltages.
Best Answer
Forget sampling rate for a few seconds... Think about sampling period for a second, which is the time interval between two consecutive samples. This time can be an integer or any real number (as long as it’s positive, of course).
Sampling rate is simply the inverse of sampling period. Does it make more sense this way?