OK, let's talk about the first computer - the ENIAC.I will concede that it plays a role in the development of magnetic memory. But bear in mind the first computers didn't have this sort of memory. So its doubtful Maxwell's equations were crucial to the actual invention of the computer.
To build an ENIAC you need logic elements. Logic elements need to exhibit two characteristics - decisionmaking and gain. Decisionmaking means you need to be able to perform logical operations - say an AND function or a NOR function. It also needs gain, so that a given signal (a bit, a clock, a number, whatever) is not lost due to losses. In ENIAC's time the only thing available to do this in any reasonable time is a vacuum tube, which can provide both.
How do you make vacuum tubes? Well, you have to figure out how to switch an electron stream on and off. How do you do this? What voltages do you need? You get those answers from the Lorentz force law. What is that based on? Maxwell's equations.